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            Lecture 19: Conditional Probability
         
    
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<p><strong>Description:</strong> Covers conditional probability and its applications to examples including medical testing, gambling, and court cases.</p>
<p><strong>Speaker:</strong> Tom Leighton</p>
<p><strong>Instructor's Note:</strong> The actual details of the Berkeley sex discrimination case may have been different than what was stated in the lecture, so it is best to consider the description given in lecture as fictional but illustrative of the mathematical point being made.</p>
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<p><span m="390">The</span> <span m="510">following</span> <span m="950">content</span> <span m="1540">is</span> <span m="1650">provided</span> <span m="2100">under</span> <span m="2380">a</span> <span m="2420">Creative</span> <span m="2820">Commons</span> <span m="3230">license.</span> <span m="4340">Your</span> <span m="4520">support</span> <span m="5030">will</span> <span m="5190">help</span> <span m="5430">MIT</span> <span m="5880">OpenCourseWare</span> <span m="6670">continue</span> <span m="7180">to</span> <span m="7270">offer</span> <span m="7680">high</span> <span m="7910">quality</span> <span m="8440">educational</span> <span m="9060">resources</span> <span m="9690">for</span> <span m="9840">free.</span> <span m="11040">To</span> <span m="11050">make</span> <span m="11250">a</span> <span m="11300">donation</span> <span m="11980">or</span> <span m="12250">view</span> <span m="12690">additional</span> <span m="13120">materials</span> <span m="13650">from</span> <span m="13800">hundreds</span> <span m="14240">of</span> <span m="14350">MIT</span> <span m="14780">courses,</span> <span m="15890">visit</span> <span m="16100">MIT</span> <span m="16520">OpenCourseWare</span> <span m="17570">at</span> <span m="17740">ocw.mit.edu.</span> </p>
<p><span m="23160">PROFESSOR: Just</span> <span m="23420">a</span> <span m="23440">reminder,</span> <span m="23990">drop</span> <span m="24350">day</span> <span m="24540">is</span> <span m="24690">tomorrow.</span> <span m="25370">So</span> <span m="25850">if</span> <span m="25980">you</span> <span m="26130">were</span> <span m="26250">thinking</span> <span m="26530">about</span> <span m="26750">dropping</span> <span m="27140">the</span> <span m="27240">course or</span> <span m="27590">in</span> <span m="28070">danger</span> <span m="28530">of</span> <span m="28630">a</span> <span m="28930">bad</span> <span m="29290">grade or</span> <span m="29610">something,</span> <span m="29960">tomorrow's</span> <span m="30390">the</span> <span m="30470">last</span> <span m="30810">chance</span> <span m="31160">to</span> <span m="31850">bail</span> <span m="32200">out.</span> <span m="34420">Last</span> <span m="34710">time</span> <span m="34860">we</span> <span m="34980">began</span> <span m="35390">our</span> <span m="35480">discussion on</span> <span m="35960">probability</span> <span m="36660">with</span> <span m="37070">the</span> <span m="37260">Monty</span> <span m="37640">Hall</span> <span m="38230">game--</span> <span m="38620">the</span> <span m="38710">Monty</span> <span m="39020">Hall</span> <span m="39210">problem.</span> <span m="40440">And</span> <span m="40550">as</span> <span m="40740">part</span> <span m="41100">of</span> <span m="41150">the</span> <span m="41290">analysis,</span> <span m="41980">we</span> <span m="42110">made</span> <span m="42440">assumptions</span> <span m="43280">of</span> <span m="43370">the</span> <span m="43470">form</span> <span m="44120">that</span> <span m="45620">given</span> <span m="45940">that</span> <span m="46150">Carol</span> <span m="46660">placed</span> <span m="47100">the</span> <span m="47220">prize</span> <span m="47590">in</span> <span m="47960">box</span> <span m="48580">1,</span> <span m="50010">the</span> <span m="50140">probability</span> <span m="50920">that</span> <span m="51060">the</span> <span m="51160">contestant</span> <span m="51880">chooses</span> <span m="52440">box</span> <span m="52830">1</span> <span m="53160">is</span> <span m="53430">1/3.</span> <span m="55280">Now,</span> <span m="56290">this</span> <span m="56520">is</span> <span m="56660">an</span> <span m="56750">example</span> <span m="57310">of</span> <span m="57390">something</span> <span m="57620">that's</span> <span m="57780">called</span> <span m="58270">a</span> <span m="58390">conditional</span> <span m="59630">probability.</span> <span m="61100">And</span> <span m="61120">that's</span> <span m="61310">what</span> <span m="61390">we're</span> <span m="61540">going</span> <span m="61660">to</span> <span m="61750">study</span> <span m="62110">today.</span> </p>
<p></p>
<p><span m="73930">Now,</span> <span m="74100">in</span> <span m="74240">general,</span> <span m="75550">you</span> <span m="75630">have</span> <span m="75900">something</span> <span m="76170">like</span> <span m="76380">the</span> <span m="76550">conditional</span> <span m="77170">probability</span> <span m="78470">that</span> <span m="78690">an</span> <span m="78810">event,</span> <span m="79690">A,</span> <span m="79950">happens</span> <span m="80880">given</span> <span m="81770">that</span> <span m="81940">some</span> <span m="82160">other</span> <span m="82550">event,</span> <span m="82930">B,</span> <span m="83760">has</span> <span m="83960">already</span> <span m="84210">taken</span> <span m="84550">place.</span> <span m="85770">And</span> <span m="85980">you</span> <span m="86150">write</span> <span m="86400">that</span> <span m="86680">down</span> <span m="88270">as</span> <span m="88480">a</span> <span m="88560">probability</span> <span m="89180">of</span> <span m="89340">A</span> <span m="90120">given</span> <span m="91880">B.</span> <span m="98650">And</span> <span m="98840">both</span> <span m="99150">A</span> <span m="99390">and B are</span> <span m="99680">events.</span> <span m="104190">Now,</span> <span m="104610">the</span> <span m="104750">example</span> <span m="105220">from</span> <span m="105360">Monty</span> <span m="105700">Hall--</span> <span m="106665">and</span> <span m="107090">actually,</span> <span m="107500">we had</span> <span m="107660">several--</span> <span m="108440">but</span> <span m="108770">you</span> <span m="108870">might</span> <span m="109090">have</span> <span m="109460">B</span> <span m="109690">being</span> <span m="110020">the</span> <span m="110170">event</span> <span m="112370">that</span> <span m="112660">Carol</span> <span m="116230">places</span> <span m="116750">the</span> <span m="116840">prize</span> <span m="118200">in</span> <span m="118650">box</span> <span m="119180">1.</span> <span m="123820">And</span> <span m="124230">A</span> <span m="124550">might</span> <span m="125000">be the</span> <span m="125140">event</span> <span m="126871">that</span> <span m="127420">the</span> <span m="127520">contestant</span> <span m="132190">chooses</span> <span m="135000">box</span> <span m="135330">1.</span> </p>
<p><span m="137300">And</span> <span m="137510">we</span> <span m="137650">assumed</span> <span m="138320">for</span> <span m="138390">the</span> <span m="138470">Monty</span> <span m="138820">Hall</span> <span m="139110">game</span> <span m="140200">that</span> <span m="145220">the</span> <span m="145380">probability</span> <span m="146700">of</span> <span m="146900">A</span> <span m="147460">given</span> <span m="147740">B</span> <span m="147950">in</span> <span m="148050">this</span> <span m="148260">case</span> <span m="149200">was</span> <span m="149360">1/3</span> <span m="149670">third</span> <span m="150750">because</span> <span m="151150">the</span> <span m="151250">contestant</span> <span m="152770">didn't</span> <span m="153110">know</span> <span m="153460">where</span> <span m="153680">the</span> <span m="154510">prize</span> <span m="154850">was.</span> <span m="157960">Now</span> <span m="158140">in</span> <span m="158270">general,</span> <span m="158850">there's</span> <span m="158920">a</span> <span m="158960">very</span> <span m="159280">simple</span> <span m="159690">formula</span> <span m="160320">to</span> <span m="160420">compute</span> <span m="161180">the</span> <span m="161290">probability</span> <span m="162120">of</span> <span m="162220">A</span> <span m="162370">given</span> <span m="162660">B.</span> <span m="164000">In</span> <span m="164500">fact,</span> <span m="164880">we'll</span> <span m="165060">treat it</span> <span m="165360">as</span> <span m="165480">a</span> <span m="165540">definition.</span> <span m="168920">Assuming</span> <span m="169420">the</span> <span m="169540">probability</span> <span m="170190">of</span> <span m="170280">B</span> <span m="170550">is</span> <span m="170720">non-zero</span> <span m="174360">than</span> <span m="174760">the</span> <span m="174850">probability</span> <span m="175700">of</span> <span m="175810">A</span> <span m="176040">given</span> <span m="176350">B</span> <span m="178470">is</span> <span m="178680">just</span> <span m="179280">the</span> <span m="180340">probability</span> <span m="181260">of</span> <span m="181836">A</span> <span m="182690">and</span> <span m="183090">B</span> <span m="183440">happening,</span> <span m="183970">both</span> <span m="184290">happening,</span> <span m="185460">divided</span> <span m="186000">by</span> <span m="186130">the</span> <span m="186250">probability</span> <span m="187860">of</span> <span m="188280">B</span> <span m="188600">happening.</span> </p>
<p><span m="190700">And</span> <span m="190870">you</span> <span m="190970">can</span> <span m="191110">see</span> <span m="191550">why</span> <span m="191840">this</span> <span m="192100">makes</span> <span m="192450">sense</span> <span m="193320">when</span> <span m="193530">the</span> <span m="193620">picture--</span> <span m="195270">say</span> <span m="195620">this</span> <span m="195850">is</span> <span m="195960">our</span> <span m="196100">sample</span> <span m="196560">space.</span> <span m="198560">And</span> <span m="198830">let</span> <span m="199040">this</span> <span m="199330">be</span> <span m="199730">the</span> <span m="199860">event,</span> <span m="200150">A,</span> <span m="201900">and</span> <span m="202100">this</span> <span m="202370">be</span> <span m="202560">the</span> <span m="202720">event,</span> <span m="203320">B.</span> <span m="205740">Now</span> <span m="205930">we're</span> <span m="206060">conditioning</span> <span m="207550">on</span> <span m="207780">the</span> <span m="207870">fact</span> <span m="208440">that</span> <span m="208540">B</span> <span m="209030">happened.</span> <span m="211110">Now</span> <span m="211300">once</span> <span m="211580">we've</span> <span m="211730">conditioned</span> <span m="212270">on</span> <span m="212430">that,</span> <span m="213360">all</span> <span m="213720">this</span> <span m="213930">stuff</span> <span m="214240">outside</span> <span m="214700">of</span> <span m="214800">B</span> <span m="214960">is</span> <span m="215100">no</span> <span m="215270">longer</span> <span m="215600">possible.</span> </p>
<p><span m="217320">All</span> <span m="217630">those</span> <span m="217920">outcomes</span> <span m="218970">are</span> <span m="219140">no</span> <span m="219320">longer</span> <span m="220650">in</span> <span m="220820">the</span> <span m="221050">space</span> <span m="221440">of</span> <span m="221520">consideration.</span> <span m="222930">The</span> <span m="223090">only</span> <span m="223380">outcomes</span> <span m="223810">left</span> <span m="224230">are in</span> <span m="224360">B.</span> <span m="225070">So</span> <span m="225240">in</span> <span m="225310">some</span> <span m="225560">sense</span> <span m="225850">we've</span> <span m="225990">shrunk</span> <span m="226460">the</span> <span m="226550">sample</span> <span m="226980">space</span> <span m="227310">to</span> <span m="227400">be</span> <span m="227530">B.</span> <span m="229050">And</span> <span m="229250">all</span> <span m="229330">we</span> <span m="229470">care</span> <span m="229800">about</span> <span m="230100">is</span> <span m="230210">the</span> <span m="230310">probability</span> <span m="230900">that</span> <span m="231140">A</span> <span m="231490">happens</span> <span m="232660">inside</span> <span m="233220">this</span> <span m="233420">new</span> <span m="233590">sample</span> <span m="233980">space.</span> <span m="235410">And</span> <span m="235560">that</span> <span m="235760">is,</span> <span m="236580">we're</span> <span m="236750">asking</span> <span m="237020">the</span> <span m="237110">probability</span> <span m="237660">1</span> <span m="237940">of</span> <span m="238090">these</span> <span m="238400">outcomes</span> <span m="238950">happens</span> <span m="239980">given</span> <span m="240350">that</span> <span m="240870">this</span> <span m="241140">is</span> <span m="241240">the</span> <span m="241330">sample</span> <span m="241730">space.</span> </p>
<p><span m="243040">Well,</span> <span m="243060">this</span> <span m="243310">is</span> <span m="243440">just</span> <span m="243780">A</span> <span m="243980">intersect</span> <span m="244530">B</span> <span m="246020">because</span> <span m="246390">you</span> <span m="246775">still</span> <span m="247160">have</span> <span m="247310">to have</span> <span m="247470">A happen,</span> <span m="247790">but</span> <span m="247920">now</span> <span m="248050">you're</span> <span m="248230">inside</span> <span m="248560">of</span> <span m="248620">B.</span> <span m="249650">And</span> <span m="249840">then</span> <span m="250030">we</span> <span m="250140">divide</span> <span m="251600">by</span> <span m="251850">probability</span> <span m="252440">of</span> <span m="252510">B.</span> <span m="253530">So</span> <span m="253710">we</span> <span m="253830">normalize</span> <span m="254660">this</span> <span m="254950">to</span> <span m="255020">be</span> <span m="255130">probability</span> <span m="255690">one.</span> </p>
<p><span m="257370">OK.</span> <span m="257649">Because</span> <span m="257839">we're</span> <span m="257959">saying</span> <span m="258399">B</span> <span m="258620">happened--</span> <span m="259490">we're</span> <span m="259620">conditioning</span> <span m="260180">on</span> <span m="260300">that.</span> <span m="260970">Therefore,</span> <span m="261459">the</span> <span m="261480">probability</span> <span m="262220">of</span> <span m="262300">these</span> <span m="262560">outcomes</span> <span m="263050">must</span> <span m="263280">be</span> <span m="263390">1.</span> <span m="264680">So</span> <span m="264890">we</span> <span m="265150">divide</span> <span m="265850">by</span> <span m="266000">the</span> <span m="266110">probability</span> <span m="266810">of</span> <span m="266890">B.</span> <span m="267480">So</span> <span m="267780">we</span> <span m="267920">normalize.</span> </p>
<p><span m="268660">This</span> <span m="268920">now</span> <span m="269080">becomes--</span> <span m="270050">the</span> <span m="270300">probability</span> <span m="270690">of</span> <span m="270750">A</span> <span m="270860">given</span> <span m="271100">B</span> <span m="271290">is</span> <span m="271490">this</span> <span m="271640">share</span> <span m="272330">of</span> <span m="272560">B</span> <span m="274100">weighted</span> <span m="274370">by</span> <span m="274500">the</span> <span m="274640">outcomes.</span> <span m="276380">OK.</span> <span m="279330">All right.</span> <span m="279520">For</span> <span m="279680">example</span> <span m="280280">then,</span> <span m="283320">what's</span> <span m="283610">the</span> <span m="283730">probability</span> <span m="284630">of</span> <span m="284720">B</span> <span m="285080">given</span> <span m="285370">B?</span> <span m="287970">what's</span> <span m="288140">that</span> <span m="288320">equal?</span> <span m="289740">1.</span> </p>
<p><span m="290700">OK.</span> <span m="291470">Because</span> <span m="291640">we</span> <span m="291720">said</span> <span m="291940">it</span> <span m="292040">happened--</span> <span m="292490">so</span> <span m="292620">it</span> <span m="292740">happens</span> <span m="293170">with</span> <span m="293260">probability</span> <span m="293720">1.</span> <span m="294406">Or,</span> <span m="294750">using</span> <span m="295020">the</span> <span m="295110">formula,</span> <span m="295590">that's</span> <span m="295820">just</span> <span m="296440">probability</span> <span m="296880">of</span> <span m="296950">B</span> <span m="297860">and</span> <span m="298200">B</span> <span m="298890">divided</span> <span m="299280">by</span> <span m="299430">probability</span> <span m="299990">of</span> <span m="300100">B.</span> <span m="301520">Well,</span> <span m="301680">that</span> <span m="301890">equals</span> <span m="303170">the</span> <span m="303270">probability</span> <span m="303880">of</span> <span m="303980">B</span> <span m="305140">divided</span> <span m="305470">by</span> <span m="305570">the</span> <span m="305680">probability</span> <span m="306170">of</span> <span m="306240">B,</span> <span m="307470">which</span> <span m="307670">is</span> <span m="307790">1.</span> </p>
<p><span m="309551">All right.</span> <span m="310010">Any</span> <span m="310180">questions</span> <span m="310700">about</span> <span m="310910">the</span> <span m="310990">definition</span> <span m="313010">of</span> <span m="313170">the</span> <span m="313770">conditional</span> <span m="314200">probability?</span> <span m="315970">Very</span> <span m="316210">simple.</span> <span m="316500">And</span> <span m="316790">it's</span> <span m="316890">easy</span> <span m="317290">to</span> <span m="317390">work</span> <span m="317680">with</span> <span m="317880">using</span> <span m="318150">the</span> <span m="318240">formulas.</span> </p>
<p><span m="320130">Now,</span> <span m="320350">there's</span> <span m="320710">a</span> <span m="321150">nice</span> <span m="321450">rule</span> <span m="321950">called</span> <span m="322230">the</span> <span m="322310">product</span> <span m="322850">rule,</span> <span m="323440">which</span> <span m="323570">follows</span> <span m="323840">from</span> <span m="324110">the</span> <span m="324180">definition</span> <span m="324940">very</span> <span m="325190">simply.</span> <span m="331600">The</span> <span m="331710">product</span> <span m="332180">rule</span> <span m="332530">says</span> <span m="333080">that</span> <span m="333880">the</span> <span m="333970">probability</span> <span m="334550">of</span> <span m="334630">A and</span> <span m="335010">B</span> <span m="335450">for</span> <span m="335560">two</span> <span m="335740">events</span> <span m="337370">is</span> <span m="337570">equal</span> <span m="337880">to</span> <span m="337950">the</span> <span m="338060">probability</span> <span m="338750">of</span> <span m="338850">B</span> <span m="340290">times</span> <span m="340990">the</span> <span m="341090">probability</span> <span m="341690">of</span> <span m="341830">A</span> <span m="342220">given</span> <span m="342500">B.</span> <span m="344880">And</span> <span m="345040">that's</span> <span m="345260">just</span> <span m="345510">follow</span> <span m="345880">straightforwardly</span> <span m="347060">from</span> <span m="347210">this</span> <span m="347410">definition.</span> </p>
<p><span m="349700">Just</span> <span m="349990">multiply</span> <span m="350620">by</span> <span m="350780">probability</span> <span m="351250">of</span> <span m="351320">B</span> <span m="351570">on</span> <span m="351660">both</span> <span m="351860">sides.</span> <span m="354120">All</span> <span m="354220">right.</span> <span m="354250">So</span> <span m="354350">now</span> <span m="354470">you</span> <span m="354570">have</span> <span m="354780">a</span> <span m="354890">rule</span> <span m="355180">of</span> <span m="355250">computing</span> <span m="355660">a</span> <span m="355710">probability</span> <span m="356440">of</span> <span m="356660">two</span> <span m="356890">events</span> <span m="357320">simultaneously</span> <span m="358120">happening.</span> <span m="359750">So</span> <span m="359880">for</span> <span m="360050">example,</span> <span m="361190">in</span> <span m="361310">the</span> <span m="361390">Monty</span> <span m="361700">Hall</span> <span m="361960">problem,</span> <span m="363460">what's</span> <span m="363820">the</span> <span m="363930">probability</span> <span m="364630">that</span> <span m="365120">Carol</span> <span m="365600">places</span> <span m="366190">the</span> <span m="366290">prize</span> <span m="366660">in</span> <span m="366760">box</span> <span m="367090">one</span> <span m="368260">and</span> <span m="368700">that's</span> <span m="368930">the</span> <span m="369020">box</span> <span m="369410">the</span> <span m="369510">contestant</span> <span m="370070">chooses?</span> <span m="371910">All right?</span> </p>
<p><span m="372340">So if</span> <span m="372420">we</span> <span m="372510">took</span> <span m="372740">A and</span> <span m="373080">B</span> <span m="373300">as</span> <span m="373430">defined</span> <span m="373860">up</span> <span m="374000">there,</span> <span m="375650">that's</span> <span m="375890">the</span> <span m="375970">probability</span> <span m="376590">that</span> <span m="377000">Carol</span> <span m="377350">places</span> <span m="377800">it in</span> <span m="377940">box</span> <span m="378260">one</span> <span m="378880">and</span> <span m="379170">the</span> <span m="379250">contestant</span> <span m="379790">chose</span> <span m="380180">it.</span> <span m="382090">Well,</span> <span m="382410">that's</span> <span m="382620">the</span> <span m="382700">probability</span> <span m="383350">that</span> <span m="383490">the</span> <span m="383570">contestant</span> <span m="384110">chooses</span> <span m="384500">it</span> <span m="384610">is</span> <span m="384710">1/3</span> <span m="386290">times</span> <span m="386810">the</span> <span m="386910">probability</span> <span m="387440">that</span> <span m="387610">Carol</span> <span m="388120">put</span> <span m="388360">it</span> <span m="388500">there,</span> <span m="388740">given</span> <span m="389060">the</span> <span m="389150">contestant</span> <span m="389670">chose</span> <span m="389960">it,</span> <span m="390110">or</span> <span m="390270">actually,</span> <span m="390490">vice</span> <span m="390740">versa,</span> <span m="393130">Is</span> <span m="393300">1/9.</span> <span m="396282">OK?</span> </p>
<p><span m="398430">And</span> <span m="398640">this</span> <span m="398810">extends</span> <span m="399460">to</span> <span m="399600">more</span> <span m="399930">events.</span> <span m="402870">It is</span> <span m="402990">called</span> <span m="403300">the</span> <span m="403390">general</span> <span m="403880">product</span> <span m="404370">rule.</span> <span m="415500">So</span> <span m="415640">if</span> <span m="415710">you</span> <span m="415830">want</span> <span m="416050">to</span> <span m="416110">compute</span> <span m="416600">the</span> <span m="416690">probability</span> <span m="418250">of</span> <span m="418690">A1</span> <span m="419640">and</span> <span m="420280">A2</span> <span m="422170">and</span> <span m="422590">all</span> <span m="422780">the</span> <span m="422850">way</span> <span m="422990">up</span> <span m="423130">to</span> <span m="423250">An,</span> <span m="425800">that's</span> <span m="426140">simply</span> <span m="426640">the</span> <span m="427070">probability</span> <span m="427900">of</span> <span m="427990">a</span> <span m="428120">1</span> <span m="428840">happening</span> <span m="429320">all</span> <span m="429470">by</span> <span m="429650">itself</span> <span m="431090">times</span> <span m="431530">the</span> <span m="431620">probability</span> <span m="432240">of</span> <span m="432430">A2</span> <span m="433970">given</span> <span m="434330">A1</span> <span m="437760">times--</span> <span m="438045">well,</span> <span m="438330">I'll do</span> <span m="438410">the</span> <span m="438910">next</span> <span m="438970">one--</span> <span m="439470">times</span> <span m="439770">the</span> <span m="439850">probability</span> <span m="440350">of</span> <span m="440480">A3</span> <span m="442050">given</span> <span m="442430">A1</span> <span m="443360">and</span> <span m="443800">A2,</span> <span m="446620">dot,</span> <span m="447090">dot</span> <span m="447560">dot,</span> <span m="449110">times,</span> <span m="449460">finally,</span> <span m="449750">the</span> <span m="449820">probability</span> <span m="450400">of</span> <span m="450570">An</span> <span m="451710">given</span> <span m="452100">all</span> <span m="452330">the</span> <span m="452470">others.</span> </p>
<p><span m="460730">So</span> <span m="460760">that</span> <span m="460890">starts</span> <span m="461230">to</span> <span m="461320">look</span> <span m="461500">a</span> <span m="461550">little</span> <span m="461810">more</span> <span m="462010">complicated.</span> <span m="463360">But</span> <span m="463480">it</span> <span m="463570">gives</span> <span m="463750">you</span> <span m="463890">a</span> <span m="463940">handy</span> <span m="464350">way</span> <span m="464590">of</span> <span m="464710">computing</span> <span m="465960">the</span> <span m="466060">probability</span> <span m="466720">that</span> <span m="466830">an</span> <span m="466930">intersection</span> <span m="467540">of</span> <span m="467630">events</span> <span m="468000">takes</span> <span m="468300">place.</span> <span m="471080">I</span> <span m="471280">do</span> <span m="471390">This</span> <span m="471590">is</span> <span m="471720">proved</span> <span m="472330">by</span> <span m="472540">induction</span> <span m="473060">on</span> <span m="473240">n,</span> <span m="473580">just</span> <span m="473940">taking</span> <span m="474180">that</span> <span m="474400">rule</span> <span m="474700">and using</span> <span m="474980">induction</span> <span m="475380">on</span> <span m="475520">n.</span> <span m="475680">It's</span> <span m="476180">not</span> <span m="476430">hard.</span> <span m="476690">But</span> <span m="476780">we</span> <span m="476900">won't</span> <span m="477140">go</span> <span m="477280">through</span> <span m="477510">it.</span> </p>
<p><span m="480060">All</span> <span m="480160">right.</span> <span m="480310">Let's</span> <span m="480600">do</span> <span m="480790">some</span> <span m="480990">examples.</span> <span m="482770">We'll</span> <span m="482810">start</span> <span m="483100">with</span> <span m="483230">an</span> <span m="483310">easy</span> <span m="483610">one.</span> <span m="487750">Say</span> <span m="487980">you're</span> <span m="488210">playing</span> <span m="488680">a</span> <span m="489420">playoff</span> <span m="490170">series</span> <span m="491670">and</span> <span m="491860">you're</span> <span m="491960">going</span> <span m="492070">to</span> <span m="492170">play</span> <span m="492480">best</span> <span m="492840">2</span> <span m="493000">out</span> <span m="493100">of</span> <span m="493180">3.</span> <span m="493430">All right.</span> <span m="495358">So</span> <span m="496090">you</span> <span m="496210">have</span> <span m="496320">a</span> <span m="496380">best</span> <span m="497450">2</span> <span m="497800">out</span> <span m="497910">of</span> <span m="498030">3</span> <span m="501370">series.</span> <span m="502210">So</span> <span m="502370">whoever</span> <span m="502670">wins</span> <span m="502940">the</span> <span m="503030">first</span> <span m="503320">two</span> <span m="503440">games,</span> <span m="504490">best</span> <span m="504860">two</span> <span m="505020">out of</span> <span m="505110">three</span> <span m="505350">wins.</span> <span m="507590">And</span> <span m="507770">say</span> <span m="507940">you're</span> <span m="508090">told</span> <span m="508580">that</span> <span m="508690">the</span> <span m="508780">probability</span> <span m="511790">of</span> <span m="511870">winning</span> <span m="512210">the</span> <span m="512299">first</span> <span m="512659">game</span> <span m="513049">is</span> <span m="513200">1/2.</span> </p>
<p><span m="524450">So</span> <span m="524770">the</span> <span m="524900">teams</span> <span m="525250">are</span> <span m="525300">matched</span> <span m="525720">50-50</span> <span m="526400">for</span> <span m="526480">the</span> <span m="526560">first</span> <span m="526880">game.</span> <span m="528560">But</span> <span m="528760">then</span> <span m="528980">you're</span> <span m="529130">told</span> <span m="529630">that</span> <span m="529750">the</span> <span m="529840">probability</span> <span m="530640">of</span> <span m="530740">winning</span> <span m="531670">a</span> <span m="531780">game</span> <span m="533020">after</span> <span m="533210">a</span> <span m="533250">victory</span> <span m="534290">is</span> <span m="534470">higher.</span> <span m="535590">It's</span> <span m="535790">2/3.</span> <span m="542000">So</span> <span m="542130">the</span> <span m="542200">probability</span> <span m="543630">of</span> <span m="543930">winning</span> <span m="549100">immediately</span> <span m="549700">after</span> <span m="550080">a</span> <span m="550120">game</span> <span m="555040">following</span> <span m="557960">a</span> <span m="558120">win</span> <span m="560540">is</span> <span m="560820">two thirds.</span> <span m="562190">And</span> <span m="562610">similarly,</span> <span m="563950">the</span> <span m="564040">probability</span> <span m="564570">of</span> <span m="564650">winning</span> <span m="565020">after</span> <span m="565340">a</span> <span m="565400">loss</span> <span m="566050">is</span> <span m="566230">1/3.</span> </p>
<p></p>
<p><span m="583610">All right.</span> <span m="583870">And the</span> <span m="583950">idea</span> <span m="584320">here</span> <span m="584690">is</span> <span m="584790">that</span> <span m="585070">you</span> <span m="585450">win</span> <span m="585630">a</span> <span m="585670">game,</span> <span m="586000">you're</span> <span m="586050">sort of</span> <span m="586360">psyched,</span> <span m="586730">you've</span> <span m="586830">got</span> <span m="586970">momentum,</span> <span m="587860">and</span> <span m="588010">going</span> <span m="588250">into</span> <span m="588430">the</span> <span m="588520">next</span> <span m="588770">day</span> <span m="589270">you're</span> <span m="589430">more</span> <span m="589640">likely</span> <span m="589960">to</span> <span m="590040">win.</span> <span m="590870">Similarly,</span> <span m="591310">if</span> <span m="591390">you</span> <span m="591510">lost</span> <span m="591990">you're</span> <span m="592080">sort of</span> <span m="592370">down</span> <span m="592730">and</span> <span m="592810">the</span> <span m="592910">other</span> <span m="593020">guy</span> <span m="593320">has a</span> <span m="593370">better</span> <span m="593610">chance</span> <span m="593910">of</span> <span m="594020">beating</span> <span m="594280">you.</span> <span m="595250">Now,</span> <span m="595400">what</span> <span m="595530">we're</span> <span m="595650">going</span> <span m="595740">to</span> <span m="595790">try</span> <span m="596000">to</span> <span m="596090">figure</span> <span m="596470">out</span> <span m="597630">is</span> <span m="598790">the</span> <span m="598930">probability</span> <span m="599870">of</span> <span m="600000">winning</span> <span m="600420">the</span> <span m="600500">series</span> <span m="601940">given</span> <span m="603060">you</span> <span m="603210">won</span> <span m="603430">the</span> <span m="603510">first</span> <span m="603790">game.</span> <span m="605690">All</span> <span m="605790">right?</span> </p>
<p><span m="605950">Now,</span> <span m="606550">conditional</span> <span m="607060">probability</span> <span m="607690">comes</span> <span m="607980">up</span> <span m="608100">in</span> <span m="608270">two</span> <span m="608490">places</span> <span m="609000">in</span> <span m="609100">this</span> <span m="609270">problem.</span> <span m="611540">Anybody</span> <span m="611910">tell</span> <span m="612040">me</span> <span m="612180">places</span> <span m="612550">where</span> <span m="612710">it's</span> <span m="612830">come</span> <span m="613010">up?</span> <span m="613900">So I got</span> <span m="614160">the</span> <span m="614240">problem</span> <span m="614520">statement</span> <span m="614920">and</span> <span m="615010">the</span> <span m="615120">that's</span> <span m="615510">the</span> <span m="615610">goal</span> <span m="615970">is to</span> <span m="616100">figure</span> <span m="616230">out</span> <span m="616300">the</span> <span m="616390">probability</span> <span m="617000">you</span> <span m="617180">win the</span> <span m="617280">series</span> <span m="617670">given</span> <span m="617870">you</span> <span m="617950">won</span> <span m="618090">the</span> <span m="618160">first</span> <span m="618450">game.</span> <span m="618680">So</span> <span m="618830">what's</span> <span m="619040">one</span> <span m="619290">place</span> <span m="619970">conditional</span> <span m="620460">probability</span> <span m="621030">is</span> <span m="621180">entering</span> <span m="622000">into</span> <span m="622240">this</span> <span m="622420">problem?</span> <span m="624880">Yeah?</span> </p>
<p><span m="626140">AUDIENCE: The</span> <span m="626470">probability</span> <span m="627110">changes</span> <span m="627550">depending</span> <span m="628010">on</span> <span m="629100">the</span> <span m="629200">result</span> <span m="629590">of</span> <span m="629870">the</span> <span m="630310">previous game.</span> </p>
<p><span m="631540">PROFESSOR: That's</span> <span m="631875">true.</span> <span m="632210">The</span> <span m="632300">probability</span> <span m="632790">of</span> <span m="632860">winning</span> <span m="633160">any</span> <span m="633290">particular</span> <span m="633770">game</span> <span m="634270">is</span> <span m="634400">influenced</span> <span m="634890">by</span> <span m="634990">the</span> <span m="635110">previous</span> <span m="635600">game.</span> <span m="635910">So</span> <span m="636020">you're using</span> <span m="636390">conditional</span> <span m="636770">probability</span> <span m="637210">there.</span> <span m="638780">All right.</span> <span m="640140">And</span> <span m="640310">where</span> <span m="640500">else?</span> <span m="641860">Yeah.</span> </p>
<p><span m="642546">AUDIENCE: [INAUDIBLE]</span> <span m="644490">you</span> <span m="644976">have</span> <span m="645462">to</span> <span m="645948">take</span> <span m="646434">into</span> <span m="646920">account</span> <span m="647406">[INAUDIBLE].</span> </p>
<p><span m="648378">PROFESSOR: That's</span> <span m="648864">interesting.</span> <span m="649350">That will be another</span> <span m="649700">question</span> <span m="650020">we're</span> <span m="650090">going</span> <span m="650180">to</span> <span m="650230">look</span> <span m="650420">at.</span> <span m="650500">What's</span> <span m="650660">the</span> <span m="650740">probability</span> <span m="651230">of</span> <span m="651880">playing</span> <span m="652190">three</span> <span m="652400">games?</span> <span m="654050">Yep.</span> <span m="655340">That's</span> <span m="655580">one.</span> </p>
<p><span m="657570">OK.</span> <span m="658020">Well,</span> <span m="659180">the</span> <span m="659300">question</span> <span m="659700">we're</span> <span m="659850">after,</span> <span m="660180">what's</span> <span m="660410">the</span> <span m="660490">probability</span> <span m="660940">of</span> <span m="660990">winning</span> <span m="661350">the</span> <span m="661430">series</span> <span m="661940">given</span> <span m="662310">that you</span> <span m="662420">won</span> <span m="662700">the</span> <span m="662790">first</span> <span m="663050">game.</span> <span m="664040">We're</span> <span m="664170">going</span> <span m="664280">to</span> <span m="664340">compute</span> <span m="664880">a</span> <span m="664940">conditional</span> <span m="665380">probability</span> <span m="665900">there.</span> <span m="667040">So</span> <span m="667110">it's</span> <span m="667220">coming</span> <span m="667500">up</span> <span m="667610">in</span> <span m="667700">a</span> <span m="667760">couple</span> <span m="668060">of</span> <span m="668140">places</span> <span m="668560">here.</span> </p>
<p><span m="669954">All right.</span> <span m="670870">Let's</span> <span m="671090">figure</span> <span m="671240">this</span> <span m="671440">out.</span> <span m="672650">It's</span> <span m="672810">easy</span> <span m="673170">to</span> <span m="673270">do</span> <span m="674150">given</span> <span m="674390">the</span> <span m="674460">tree</span> <span m="674700">method.</span> <span m="676710">So</span> <span m="676880">let's</span> <span m="677150">make</span> <span m="678330">the</span> <span m="678430">tree</span> <span m="678960">for</span> <span m="679070">this.</span> <span m="679610">So</span> <span m="679830">we</span> <span m="679980">have</span> <span m="681240">possibly</span> <span m="681740">three</span> <span m="682000">games</span> <span m="682460">there's</span> <span m="682640">game</span> <span m="682880">one,</span> <span m="684590">game</span> <span m="684920">two,</span> <span m="687300">and</span> <span m="687430">game</span> <span m="687690">three.</span> </p>
<p><span m="691060">Game</span> <span m="691260">one,</span> <span m="691920">you</span> <span m="692030">can</span> <span m="692220">win</span> <span m="692370">or</span> <span m="692470">lose.</span> <span m="692910">There's</span> <span m="693090">two</span> <span m="694310">branches.</span> <span m="696560">Game</span> <span m="696880">two</span> <span m="697240">you</span> <span m="697390">can</span> <span m="697560">win</span> <span m="697720">or</span> <span m="697800">lose.</span> <span m="701350">And</span> <span m="701550">now,</span> <span m="701710">game</span> <span m="702070">three--</span> <span m="703660">well,</span> <span m="703900">it doesn't</span> <span m="704210">even</span> <span m="704330">take</span> <span m="704530">place</span> <span m="705160">here.</span> </p>
<p><span m="706650">But</span> <span m="706790">it</span> <span m="706900">does</span> <span m="707200">here.</span> <span m="707450">You</span> <span m="707550">can</span> <span m="707700">win</span> <span m="707830">or</span> <span m="707910">lose</span> <span m="708200">here.</span> <span m="710800">And</span> <span m="711120">you</span> <span m="711230">could</span> <span m="711760">win</span> <span m="711950">or</span> <span m="712050">lose</span> <span m="712340">here.</span> <span m="714520">And</span> <span m="714720">here</span> <span m="715090">the</span> <span m="715250">series</span> <span m="715600">is</span> <span m="715740">over.</span> <span m="716120">So there</span> <span m="716340">is</span> <span m="716480">no</span> <span m="716640">game</span> <span m="716910">three</span> <span m="717180">in</span> <span m="717290">that</span> <span m="717460">case.</span> </p>
<p><span m="720040">The</span> <span m="720270">probabilities</span> <span m="721150">are</span> <span m="721220">next</span> <span m="721620">we</span> <span m="721730">put</span> <span m="722420">a</span> <span m="722610">probability</span> <span m="723030">of</span> <span m="723110">every</span> <span m="723320">branch</span> <span m="723740">here.</span> <span m="724900">Game</span> <span m="725140">one</span> <span m="725300">is</span> <span m="725430">50-50.</span> <span m="728380">What's</span> <span m="728610">the</span> <span m="728700">probability</span> <span m="729005">you</span> <span m="729310">take</span> <span m="729560">this</span> <span m="729770">branch?</span> <span m="733530">2/3,</span> <span m="734380">because</span> <span m="734560">you're</span> <span m="734750">on</span> <span m="734940">the</span> <span m="735030">path</span> <span m="735450">where</span> <span m="735620">you won</span> <span m="735790">the</span> <span m="735870">first</span> <span m="736180">game.</span> </p>
<p><span m="737170">You</span> <span m="737310">win</span> <span m="737420">the</span> <span m="737510">second</span> <span m="737850">game</span> <span m="738030">with</span> <span m="738150">2/3.</span> <span m="738760">You</span> <span m="738900">lose</span> <span m="739230">with</span> <span m="739380">1/3.</span> <span m="741540">Now</span> <span m="741680">here</span> <span m="742100">you're</span> <span m="742270">on</span> <span m="742370">the</span> <span m="742460">path</span> <span m="742910">where you</span> <span m="743050">lost</span> <span m="743490">the</span> <span m="743560">first</span> <span m="743880">game.</span> <span m="745020">So</span> <span m="745200">this</span> <span m="745560">has</span> <span m="745790">1/3</span> <span m="747210">and</span> <span m="747360">this</span> <span m="747560">has</span> <span m="747780">2/3.</span> </p>
<p><span m="750260">All</span> <span m="750360">right?</span> <span m="750430">And</span> <span m="750510">then</span> <span m="750610">lastly,</span> <span m="750980">what's</span> <span m="751200">the</span> <span m="751290">probability</span> <span m="751930">I</span> <span m="752010">have</span> <span m="752510">the</span> <span m="752760">win</span> <span m="753850">on</span> <span m="754120">the</span> <span m="756260">third</span> <span m="756610">game</span> <span m="756920">here?</span> <span m="759060">1/3,</span> <span m="759740">because</span> <span m="759920">I</span> <span m="760010">just</span> <span m="760450">lost</span> <span m="760900">the</span> <span m="760970">last</span> <span m="761350">game.</span> <span m="762150">That's</span> <span m="762510">all</span> <span m="762740">I'm</span> <span m="763040">conditioning</span> <span m="763560">on.</span> </p>
<p><span m="764220">So</span> <span m="764340">that</span> <span m="764520">becomes</span> <span m="764840">1/3.</span> <span m="766250">And</span> <span m="766400">this</span> <span m="766600">is</span> <span m="766740">2/3</span> <span m="767330">now.</span> <span m="769150">And</span> <span m="769240">then</span> <span m="769370">here</span> <span m="769690">I</span> <span m="769770">just</span> <span m="770220">won</span> <span m="770430">a</span> <span m="770490">game.</span> <span m="770770">So</span> <span m="770880">I've</span> <span m="770980">got</span> <span m="771200">2/3</span> <span m="772450">and</span> <span m="772600">1/3.</span> </p>
<p><span m="774386">All right.</span> <span m="775290">So</span> <span m="775420">I</span> <span m="775460">got</span> <span m="775690">all</span> <span m="775810">the</span> <span m="775910">probabilities.</span> <span m="777890">And</span> <span m="778060">now</span> <span m="778270">I</span> <span m="778360">need</span> <span m="778550">to</span> <span m="778620">figure</span> <span m="779060">out</span> <span m="780310">for</span> <span m="780460">the</span> <span m="780620">sample</span> <span m="781170">points</span> <span m="782560">what's</span> <span m="782870">their</span> <span m="783050">probability.</span> <span m="783730">So</span> <span m="784020">this</span> <span m="784270">sample</span> <span m="784600">point</span> <span m="785100">we'll</span> <span m="785230">call</span> <span m="785610">win-win.</span> </p>
<p><span m="787760">This</span> <span m="788000">sample</span> <span m="788340">point</span> <span m="789650">is</span> <span m="790690">win-lose-win.</span> <span m="795320">This</span> <span m="795520">one's</span> <span m="795740">win-lose-lose.</span> <span m="800130">Then</span> <span m="800360">we</span> <span m="800480">have</span> <span m="801000">lose-win-win,</span> <span m="804650">lose-win-lose,</span> <span m="807780">and</span> <span m="807920">then</span> <span m="808150">lose-lose.</span> </p>
<p><span m="810040">So</span> <span m="810130">I</span> <span m="810180">got</span> <span m="810360">six</span> <span m="810690">sample</span> <span m="811100">points.</span> <span m="812010">And</span> <span m="812100">let's</span> <span m="812330">figure</span> <span m="812650">out</span> <span m="812740">the</span> <span m="812830">probability</span> <span m="813460">for</span> <span m="813660">each</span> <span m="813820">one.</span> <span m="815430">Now</span> <span m="816400">remember</span> <span m="816680">the</span> <span m="816820">rule</span> <span m="817020">we</span> <span m="817110">had</span> <span m="817320">for</span> <span m="817390">the</span> <span m="817480">tree</span> <span m="817750">method.</span> <span m="819280">I</span> <span m="819400">just</span> <span m="819590">multiply</span> <span m="820220">these</span> <span m="820470">things.</span> </p>
<p><span m="822570">Well,</span> <span m="822850">in</span> <span m="822930">fact,</span> <span m="823380">the</span> <span m="823390">reason</span> <span m="823720">we</span> <span m="823840">have</span> <span m="824090">that</span> <span m="824310">rule</span> <span m="824820">is</span> <span m="824910">because</span> <span m="825230">that</span> <span m="825480">is</span> <span m="825690">the</span> <span m="825800">same</span> <span m="826210">as</span> <span m="826310">the</span> <span m="826420">product</span> <span m="826900">rule.</span> <span m="829120">Because</span> <span m="829360">what</span> <span m="829500">I'm</span> <span m="829650">asking</span> <span m="830060">here</span> <span m="830440">to</span> <span m="830510">compute</span> <span m="830830">the</span> <span m="830930">probability</span> <span m="831530">of</span> <span m="831600">this</span> <span m="831820">guy</span> <span m="834050">is--</span> <span m="836141">so</span> <span m="836580">the</span> <span m="836700">product</span> <span m="837190">rule</span> <span m="842850">gives</span> <span m="843210">the</span> <span m="843310">probability</span> <span m="845230">of</span> <span m="845390">a</span> <span m="845480">win-win</span> <span m="846350">scenario--</span> <span m="847030">win</span> <span m="847220">the</span> <span m="847300">first</span> <span m="847550">game,</span> <span m="847700">win</span> <span m="847820">the</span> <span m="847900">second</span> <span m="848290">game.</span> <span m="849420">By</span> <span m="849580">the</span> <span m="849700">product</span> <span m="850240">rule</span> <span m="850480">is</span> <span m="850550">the</span> <span m="850650">probability</span> <span m="851210">that</span> <span m="851340">I</span> <span m="851430">win</span> <span m="851830">the</span> <span m="851910">first</span> <span m="852250">game</span> <span m="855390">times</span> <span m="855850">the</span> <span m="855940">probability</span> <span m="856580">that</span> <span m="856760">I</span> <span m="857270">win</span> <span m="857590">the</span> <span m="857680">second</span> <span m="858140">game</span> <span m="859560">given</span> <span m="859950">that</span> <span m="860080">I</span> <span m="860150">won</span> <span m="860800">the</span> <span m="860900">first</span> <span m="861230">game.</span> <span m="863570">That's</span> <span m="863760">what</span> <span m="863860">the</span> <span m="863940">product</span> <span m="864330">rule</span> <span m="864550">says.</span> </p>
<p><span m="867280">Probability</span> <span m="868000">I win</span> <span m="868220">the</span> <span m="868310">first</span> <span m="868670">game</span> <span m="868940">is</span> <span m="869050">1/2</span> <span m="871310">times</span> <span m="871590">the</span> <span m="871680">probability</span> <span m="872220">I win</span> <span m="872570">the</span> <span m="872670">second</span> <span m="873150">given</span> <span m="873480">that</span> <span m="873590">I</span> <span m="873640">won</span> <span m="873860">the</span> <span m="873950">first</span> <span m="874540">is</span> <span m="875210">2/3.</span> <span m="877590">So</span> <span m="877720">that</span> <span m="877960">equals</span> <span m="879850">1/3.</span> <span m="885790">So</span> <span m="885910">what</span> <span m="886070">we're</span> <span m="886160">doing</span> <span m="886540">here</span> <span m="886750">now</span> <span m="886970">is</span> <span m="887090">giving</span> <span m="887310">you</span> <span m="887390">the</span> <span m="887510">formal</span> <span m="888030">justification</span> <span m="888930">for</span> <span m="889030">that</span> <span m="889330">rule</span> <span m="889770">that we</span> <span m="889890">had</span> <span m="890090">last</span> <span m="890440">time</span> <span m="891270">and</span> <span m="891320">that</span> <span m="891400">you'll</span> <span m="891610">always</span> <span m="892040">use--</span> <span m="892305">is</span> <span m="892570">the</span> <span m="892670">probability</span> <span m="893250">of</span> <span m="893310">a</span> <span m="893360">sample</span> <span m="893770">point</span> <span m="894580">is</span> <span m="894750">the</span> <span m="894870">product</span> <span m="895480">of</span> <span m="895540">the</span> <span m="895630">probabilities</span> <span m="896320">on</span> <span m="896440">the</span> <span m="896580">edges</span> <span m="897290">leading</span> <span m="897580">to</span> <span m="897770">it.</span> <span m="899100">It's</span> <span m="899310">just</span> <span m="899530">the</span> <span m="899620">product</span> <span m="900060">rule.</span> </p>
<p><span m="900310">Now</span> <span m="900740">the</span> <span m="900840">next</span> <span m="901130">example is</span> <span m="901630">this</span> <span m="901840">one.</span> <span m="903260">And</span> <span m="903410">here</span> <span m="903660">we're</span> <span m="903790">going</span> <span m="903900">to</span> <span m="903940">use</span> <span m="904300">the</span> <span m="904410">general</span> <span m="904920">product</span> <span m="905380">rule</span> <span m="905850">to get it.</span> <span m="909190">The</span> <span m="909310">probability</span> <span m="910930">of</span> <span m="911210">win-lose-win</span> <span m="914780">by</span> <span m="914890">the</span> <span m="915020">general</span> <span m="915400">product</span> <span m="915860">rule</span> <span m="916160">is</span> <span m="916290">the</span> <span m="916380">probability</span> <span m="917086">that</span> <span m="917770">you</span> <span m="918030">win</span> <span m="918280">the</span> <span m="918360">first</span> <span m="918780">game</span> <span m="920440">times</span> <span m="921060">the</span> <span m="921140">probability</span> <span m="921840">you</span> <span m="921990">lose</span> <span m="922750">the</span> <span m="922860">second</span> <span m="923300">game</span> <span m="924510">given</span> <span m="924820">the</span> <span m="924920">that you win</span> <span m="925400">the</span> <span m="925480">first</span> <span m="928430">times</span> <span m="928840">the</span> <span m="928930">probability</span> <span m="929660">you</span> <span m="930640">win</span> <span m="931080">the</span> <span m="931190">third</span> <span m="933590">given</span> <span m="933900">what?</span> </p>
<p><span m="934786">What</span> <span m="935230">am</span> <span m="935340">I</span> <span m="935410">given</span> <span m="935730">on</span> <span m="935830">the</span> <span m="935910">product</span> <span m="936340">rule?</span> <span m="938740">Won</span> <span m="939100">the</span> <span m="939200">first,</span> <span m="940870">lost</span> <span m="941400">the</span> <span m="941450">second.</span> <span m="943582">All right.</span> <span m="944440">Well,</span> <span m="944850">now</span> <span m="945020">we</span> <span m="945120">can</span> <span m="945270">fill</span> <span m="945500">in the</span> <span m="945570">numbers.</span> </p>
<p><span m="945960">The</span> <span m="946010">probability</span> <span m="946280">I</span> <span m="946500">win</span> <span m="946700">the</span> <span m="946790">first</span> <span m="947200">is</span> <span m="947310">a</span> <span m="947360">1/2.</span> <span m="949768">The</span> <span m="950230">probability</span> <span m="950750">that I</span> <span m="950800">lose</span> <span m="951340">the</span> <span m="951450">second</span> <span m="952740">given</span> <span m="952950">that</span> <span m="953070">I</span> <span m="953140">won</span> <span m="953550">the</span> <span m="953640">first,</span> <span m="955230">that's</span> <span m="955430">1/3.</span> <span m="957720">And</span> <span m="957800">then</span> <span m="958140">this</span> <span m="958370">one</span> <span m="958530">here,</span> <span m="958850">the</span> <span m="958960">probability</span> <span m="959440">that</span> <span m="959610">I win</span> <span m="959970">the</span> <span m="960070">third</span> <span m="960570">given</span> <span m="960780">that</span> <span m="960900">I</span> <span m="960950">won</span> <span m="961270">the</span> <span m="961360">first</span> <span m="961680">and</span> <span m="961740">lost</span> <span m="962090">the</span> <span m="962130">second,</span> <span m="963580">that</span> <span m="963890">simplifies</span> <span m="964600">the</span> <span m="964720">probability</span> <span m="965210">I</span> <span m="965250">win</span> <span m="965510">the</span> <span m="965610">third</span> <span m="967810">given</span> <span m="968040">that</span> <span m="968160">I</span> <span m="968210">lost</span> <span m="968610">the</span> <span m="968660">second.</span> <span m="969160">Doesn't</span> <span m="969450">matter</span> <span m="969800">what</span> <span m="970160">happened</span> <span m="970330">on</span> <span m="970430">the</span> <span m="970510">first.</span> <span m="971930">And</span> <span m="972140">that's</span> <span m="972350">1/3.</span> </p>
<p><span m="976010">So</span> <span m="976390">this</span> <span m="976650">is</span> <span m="977840">1/2</span> <span m="978330">times</span> <span m="978540">1/3</span> <span m="978910">times</span> <span m="979070">1/3</span> <span m="979500">is</span> <span m="979660">118.</span> <span m="984650">And</span> <span m="984950">that's</span> <span m="986020">1/18.</span> <span m="988090">And</span> <span m="988260">it's</span> <span m="988410">just</span> <span m="988610">the</span> <span m="988700">product</span> <span m="990520">because</span> <span m="990780">the</span> <span m="990870">product</span> <span m="991260">rule</span> <span m="991480">saying</span> <span m="993110">product</span> <span m="993550">of</span> <span m="993610">the</span> <span m="993690">first</span> <span m="994310">probability</span> <span m="995290">times</span> <span m="996120">this</span> <span m="996460">one,</span> <span m="996620">which</span> <span m="996810">is</span> <span m="996920">the</span> <span m="997010">conditional</span> <span m="997460">probability</span> <span m="997910">of</span> <span m="997990">being</span> <span m="998290">here</span> <span m="999170">times</span> <span m="999520">this</span> <span m="999770">one,</span> <span m="999900">which</span> <span m="1000080">is</span> <span m="1000200">a</span> <span m="1000270">conditional</span> <span m="1000680">probability</span> <span m="1001200">if</span> <span m="1001560">these</span> <span m="1001880">events</span> <span m="1003060">happened</span> <span m="1003420">before.</span> </p>
<p><span m="1005600">Any</span> <span m="1005790">questions</span> <span m="1006310">about</span> <span m="1007340">that?</span> <span m="1009180">Very</span> <span m="1009460">simple</span> <span m="1010090">to</span> <span m="1010220">do,</span> <span m="1010660">which</span> <span m="1010880">is</span> <span m="1011060">good.</span> <span m="1011270">Yeah.</span> <span m="1011740">Is there a</span> <span m="1012180">question?</span> <span m="1013610">OK.</span> <span m="1014090">All</span> <span m="1014570">right.</span> <span m="1014870">So</span> <span m="1014970">let's</span> <span m="1015020">fill</span> <span m="1015230">in</span> <span m="1015290">the</span> <span m="1015410">other</span> <span m="1015790">probabilities</span> <span m="1016380">here.</span> </p>
<p><span m="1016630">I</span> <span m="1016670">got</span> <span m="1016830">1/2,</span> <span m="1017480">1/3,</span> <span m="1017870">and</span> <span m="1018000">2/3.</span> <span m="1019500">That's</span> <span m="1019750">1/9.</span> <span m="1021820">Same</span> <span m="1022160">thing</span> <span m="1022350">here</span> <span m="1022610">is</span> <span m="1022700">1/9.</span> <span m="1024650">This</span> <span m="1024819">is</span> <span m="1024940">1/18</span> <span m="1027319">and</span> <span m="1028569">1/3.</span> <span m="1032130">OK.</span> <span m="1032569">So</span> <span m="1032730">those</span> <span m="1033000">are</span> <span m="1033030">the</span> <span m="1033130">probabilities</span> <span m="1033710">in</span> <span m="1033750">the</span> <span m="1033829">sample</span> <span m="1034210">points.</span> </p>
<p><span m="1035319">Now,</span> <span m="1035450">to</span> <span m="1035540">compute</span> <span m="1036099">the</span> <span m="1036190">probability</span> <span m="1037800">of</span> <span m="1037930">winning</span> <span m="1038170">the</span> <span m="1038250">series</span> <span m="1039780">given</span> <span m="1040119">that</span> <span m="1040200">we</span> <span m="1040300">won</span> <span m="1040510">the</span> <span m="1040589">first</span> <span m="1040940">game,</span> <span m="1041349">let's</span> <span m="1041579">define</span> <span m="1042010">the</span> <span m="1042140">events</span> <span m="1042819">here.</span> <span m="1046890">So</span> <span m="1047310">A</span> <span m="1047720">be</span> <span m="1047890">the</span> <span m="1048020">event</span> <span m="1051190">that</span> <span m="1051590">we</span> <span m="1051710">win</span> <span m="1051960">the</span> <span m="1052050">series.</span> <span m="1056220">B</span> <span m="1056570">will</span> <span m="1056720">be</span> <span m="1056910">the</span> <span m="1057050">event</span> <span m="1059760">that</span> <span m="1059930">we</span> <span m="1060080">win</span> <span m="1060560">the</span> <span m="1061660">first</span> <span m="1061980">game.</span> <span m="1065580">And</span> <span m="1065750">I</span> <span m="1065790">want</span> <span m="1066010">to</span> <span m="1066070">compute</span> <span m="1066550">the</span> <span m="1066640">probability</span> <span m="1067750">of</span> <span m="1067940">A</span> <span m="1068180">given</span> <span m="1068470">B.</span> </p>
<p><span m="1070560">And</span> <span m="1070720">we</span> <span m="1070810">use</span> <span m="1071020">our</span> <span m="1071120">formula.</span> <span m="1073020">Where's</span> <span m="1073390">the</span> <span m="1073480">formula</span> <span m="1074340">for</span> <span m="1074480">that?</span> <span m="1074825">It's</span> <span m="1075170">way</span> <span m="1075400">back</span> <span m="1075660">over</span> <span m="1075840">there.</span> <span m="1076170">The</span> <span m="1076270">probability</span> <span m="1076960">of</span> <span m="1077120">A</span> <span m="1077360">given</span> <span m="1077670">B</span> <span m="1078450">is</span> <span m="1078600">the</span> <span m="1078700">probability</span> <span m="1079210">of</span> <span m="1079290">both</span> <span m="1081200">happening,</span> <span m="1082320">the</span> <span m="1082670">probability</span> <span m="1083110">of</span> <span m="1083180">A and</span> <span m="1083630">B</span> <span m="1084710">divided</span> <span m="1085190">by</span> <span m="1085320">the</span> <span m="1085430">probability</span> <span m="1085990">of</span> <span m="1086060">B.</span> </p>
<p><span m="1089450">So</span> <span m="1089660">now</span> <span m="1089850">I</span> <span m="1089930">just</span> <span m="1090110">have</span> <span m="1090190">to</span> <span m="1090270">compute</span> <span m="1091080">these</span> <span m="1091280">probabilities.</span> <span m="1093620">So</span> <span m="1093670">to</span> <span m="1093780">do</span> <span m="1094000">that</span> <span m="1094900">I</span> <span m="1094980">got</span> <span m="1095110">to</span> <span m="1095180">figure</span> <span m="1095480">out</span> <span m="1095620">which</span> <span m="1095810">sample</span> <span m="1096230">points</span> <span m="1096590">are</span> <span m="1096700">in</span> <span m="1096820">A and</span> <span m="1097230">B</span> <span m="1098530">here.</span> <span m="1100820">So</span> <span m="1100940">let's</span> <span m="1101090">write</span> <span m="1101320">that</span> <span m="1101580">down.</span> <span m="1102030">There's</span> <span m="1102280">A,</span> <span m="1103330">B,</span> <span m="1104290">A</span> <span m="1104710">and</span> <span m="1105320">B.</span> <span m="1107340">All</span> <span m="1107490">right.</span> <span m="1107750">So</span> <span m="1109950">A</span> <span m="1110390">is</span> <span m="1110600">the</span> <span m="1110750">event</span> <span m="1111620">that</span> <span m="1111750">we</span> <span m="1111880">win</span> <span m="1112180">the</span> <span m="1112280">series.</span> <span m="1115670">Now</span> <span m="1115840">this</span> <span m="1116060">sample</span> <span m="1116390">point</span> <span m="1116640">qualifies,</span> <span m="1118630">that</span> <span m="1118880">one</span> <span m="1119060">does,</span> <span m="1120060">and</span> <span m="1120190">this</span> <span m="1120400">one.</span> </p>
<p><span m="1122910">B</span> <span m="1123160">is</span> <span m="1123270">the event</span> <span m="1123590">we</span> <span m="1123700">won</span> <span m="1123880">the</span> <span m="1123960">first</span> <span m="1124260">game.</span> <span m="1124530">And</span> <span m="1124600">that's</span> <span m="1124820">these</span> <span m="1125060">three</span> <span m="1125440">sample</span> <span m="1125820">points.</span> <span m="1128360">And</span> <span m="1128530">then</span> <span m="1129355">A</span> <span m="1129700">and</span> <span m="1130350">B</span> <span m="1131470">intersect</span> <span m="1131960">B</span> <span m="1132100">is</span> <span m="1132240">these</span> <span m="1132460">two.</span> <span m="1134690">All</span> <span m="1134780">right.</span> </p>
<p><span m="1135020">So</span> <span m="1135150">for</span> <span m="1135280">each</span> <span m="1135500">event</span> <span m="1135760">that</span> <span m="1135870">I</span> <span m="1135930">care</span> <span m="1136290">about</span> <span m="1136560">I</span> <span m="1136630">figure</span> <span m="1136860">out</span> <span m="1136980">which</span> <span m="1137210">sample</span> <span m="1137560">points</span> <span m="1137920">are</span> <span m="1137990">in</span> <span m="1138070">that</span> <span m="1138240">event.</span> <span m="1141370">And</span> <span m="1141500">now</span> <span m="1141710">I</span> <span m="1141800">just</span> <span m="1142030">add</span> <span m="1142230">the</span> <span m="1142310">probabilities</span> <span m="1142970">up.</span> <span m="1145570">So</span> <span m="1145800">what's</span> <span m="1146030">the</span> <span m="1146130">probability</span> <span m="1147380">of</span> <span m="1147500">A</span> <span m="1147940">and</span> <span m="1148360">B?</span> <span m="1157400">7/18.</span> <span m="1159070">1/3</span> <span m="1159430">plus</span> <span m="1159660">1/18.</span> </p>
<p><span m="1164810">What's</span> <span m="1165100">the</span> <span m="1165190">probability</span> <span m="1165800">of</span> <span m="1165870">B?</span> <span m="1170830">Yeah.</span> <span m="1171160">1/2,</span> <span m="1171520">9/18.</span> <span m="1172330">I</span> <span m="1172430">got</span> <span m="1172610">these</span> <span m="1172840">three</span> <span m="1173050">points.</span> <span m="1177070">So</span> <span m="1177090">this'll</span> <span m="1177400">be</span> <span m="1178410">1/3</span> <span m="1178510">third</span> <span m="1179350">plus</span> <span m="1179730">1/18</span> <span m="1181150">plus</span> <span m="1181590">the</span> <span m="1181740">extra</span> <span m="1182040">one,</span> <span m="1182210">1/9.</span> <span m="1183770">So</span> <span m="1183890">I've</span> <span m="1184010">got</span> <span m="1184560">7/18</span> <span m="1186180">over</span> <span m="1186460">9/18.</span> <span m="1189340">7/9</span> <span m="1191060">is</span> <span m="1191200">the</span> <span m="1191320">answer.</span> <span m="1192640">So</span> <span m="1192770">the</span> <span m="1192860">probability</span> <span m="1193450">we</span> <span m="1193590">win</span> <span m="1193890">the</span> <span m="1194030">Series</span> <span m="1194470">given</span> <span m="1194680">we</span> <span m="1194790">won</span> <span m="1194990">the</span> <span m="1195110">first</span> <span m="1195560">game</span> <span m="1195820">is</span> <span m="1195910">7/9.</span> <span m="1199200">Any</span> <span m="1199350">questions?</span> </p>
<p><span m="1203050">We're</span> <span m="1203200">going</span> <span m="1203310">to</span> <span m="1203400">do</span> <span m="1203570">this</span> <span m="1203730">same</span> <span m="1204180">thing</span> <span m="1204470">about</span> <span m="1204800">10</span> <span m="1205000">different</span> <span m="1205290">times.</span> <span m="1206090">OK?</span> <span m="1206850">And</span> <span m="1207050">it will</span> <span m="1207220">look</span> <span m="1207360">a</span> <span m="1207420">little</span> <span m="1207670">different</span> <span m="1207970">each</span> <span m="1208150">time</span> <span m="1208460">maybe.</span> <span m="1209170">But</span> <span m="1209380">it's</span> <span m="1209490">the</span> <span m="1209580">same</span> <span m="1209920">idea.</span> <span m="1210250">And</span> <span m="1210350">the</span> <span m="1210410">beauty</span> <span m="1210800">here</span> <span m="1211540">is</span> <span m="1211720">it's</span> <span m="1211870">really</span> <span m="1212200">easy</span> <span m="1212440">to</span> <span m="1212540">do.</span> <span m="1213780">I'm</span> <span m="1213790">going</span> <span m="1213890">to</span> <span m="1213930">give</span> <span m="1214140">you</span> <span m="1214240">a</span> <span m="1214320">lot</span> <span m="1214530">of</span> <span m="1214580">confusing</span> <span m="1215070">examples.</span> <span m="1216400">But</span> <span m="1216580">really,</span> <span m="1216860">if</span> <span m="1216930">you</span> <span m="1217010">just</span> <span m="1217280">do</span> <span m="1217400">this</span> <span m="1217730">is</span> <span m="1218770">it's</span> <span m="1218920">going</span> <span m="1219010">to</span> <span m="1219060">be</span> <span m="1219170">very</span> <span m="1219500">easy.</span> </p>
<p><span m="1220696">All right.</span> <span m="1221220">Somebody</span> <span m="1221610">talked</span> <span m="1221900">about</span> <span m="1223160">the</span> <span m="1223320">series</span> <span m="1223690">lasting</span> <span m="1224100">three</span> <span m="1224290">games.</span> <span m="1224670">What's</span> <span m="1224910">the</span> <span m="1225000">probability</span> <span m="1225520">the</span> <span m="1225630">series</span> <span m="1226050">lasts</span> <span m="1226370">three</span> <span m="1226570">games?</span> <span m="1227722">Can</span> <span m="1228070">anybody</span> <span m="1228430">look</span> <span m="1228590">at</span> <span m="1228660">that</span> <span m="1228820">and</span> <span m="1228910">tell</span> <span m="1229250">me?</span> <span m="1233410">1/3</span> <span m="1234750">because</span> <span m="1235070">what</span> <span m="1235390">you</span> <span m="1235710">would do</span> <span m="1235940">is</span> <span m="1236070">add</span> <span m="1236250">up</span> <span m="1236410">these</span> <span m="1236680">three</span> <span m="1236920">sample</span> <span m="1237310">points.</span> <span m="1239100">And</span> <span m="1239280">it's</span> <span m="1239420">the</span> <span m="1239530">opposite</span> <span m="1239940">of</span> <span m="1240060">these</span> <span m="1240300">two.</span> <span m="1241910">So</span> <span m="1242110">it's</span> <span m="1242310">2/3</span> <span m="1242790">chance</span> <span m="1243100">of</span> <span m="1243190">two</span> <span m="1243360">games,</span> <span m="1243940">a</span> <span m="1244220">1/3</span> <span m="1244730">chance</span> <span m="1245040">of</span> <span m="1245130">three</span> <span m="1245340">games.</span> <span m="1245690">So</span> <span m="1245790">it's</span> <span m="1245930">not</span> <span m="1246270">likely</span> <span m="1246570">to</span> <span m="1246640">go</span> <span m="1246770">three</span> <span m="1246990">games.</span> </p>
<p><span m="1249910">All</span> <span m="1250040">right.</span> <span m="1251640">So</span> <span m="1252450">to</span> <span m="1252700">this</span> <span m="1252910">point,</span> <span m="1253320">we've</span> <span m="1253360">seen</span> <span m="1253570">examples</span> <span m="1254860">of</span> <span m="1255580">a</span> <span m="1255660">conditional</span> <span m="1256110">probability</span> <span m="1256660">where</span> <span m="1256830">it's</span> <span m="1257020">A</span> <span m="1257250">given</span> <span m="1257550">B</span> <span m="1257910">where</span> <span m="1258660">A</span> <span m="1259070">follows</span> <span m="1259960">B,</span> <span m="1260920">like,</span> <span m="1261220">we're</span> <span m="1261350">told</span> <span m="1261860">B</span> <span m="1262090">happened.</span> <span m="1263570">Now</span> <span m="1263780">what's</span> <span m="1263960">the</span> <span m="1264050">chance</span> <span m="1264340">of</span> <span m="1264450">A.</span> <span m="1264740">And</span> <span m="1265030">A is</span> <span m="1265170">coming</span> <span m="1265430">later.</span> <span m="1267370">The</span> <span m="1267450">probability</span> <span m="1268000">of</span> <span m="1268100">winning</span> <span m="1269100">today's</span> <span m="1269560">game</span> <span m="1269990">given</span> <span m="1270270">that</span> <span m="1270380">you</span> <span m="1270500">won</span> <span m="1270740">yesterday's</span> <span m="1271310">game,</span> <span m="1272420">the</span> <span m="1272430">probability</span> <span m="1272950">of</span> <span m="1273010">winning</span> <span m="1273240">the</span> <span m="1273320">series</span> <span m="1273890">given</span> <span m="1274120">you</span> <span m="1274230">already</span> <span m="1275010">won</span> <span m="1275250">the</span> <span m="1275330">first</span> <span m="1275590">game.</span> </p>
<p><span m="1277770">Next,</span> <span m="1278200">we're</span> <span m="1278290">going</span> <span m="1278380">to</span> <span m="1278440">look</span> <span m="1278690">at</span> <span m="1279220">the</span> <span m="1279440">opposite</span> <span m="1279920">scenario</span> <span m="1281140">where</span> <span m="1281340">the</span> <span m="1281520">events</span> <span m="1281970">are</span> <span m="1282150">reversed</span> <span m="1282600">in</span> <span m="1282680">order.</span> <span m="1284320">The</span> <span m="1284450">probability</span> <span m="1286060">that</span> <span m="1286230">you</span> <span m="1286350">won</span> <span m="1286720">the</span> <span m="1286820">first</span> <span m="1287280">game</span> <span m="1288240">given</span> <span m="1288590">that</span> <span m="1288650">you</span> <span m="1288870">won</span> <span m="1289410">the</span> <span m="1289530">series.</span> <span m="1292294">All</span> <span m="1292780">right.</span> <span m="1293120">Now,</span> <span m="1294120">this</span> <span m="1294450">is</span> <span m="1294600">inherently</span> <span m="1295540">confusing</span> <span m="1297140">because</span> <span m="1298810">if</span> <span m="1299000">you're</span> <span m="1299140">trying</span> <span m="1299470">to</span> <span m="1299560">figure--</span> <span m="1300170">if</span> <span m="1300720">you</span> <span m="1301160">know you</span> <span m="1301340">the</span> <span m="1301430">series,</span> <span m="1303130">well,</span> <span m="1303360">you</span> <span m="1303450">already</span> <span m="1303730">know</span> <span m="1304110">what</span> <span m="1304270">happened</span> <span m="1304670">in</span> <span m="1304740">the</span> <span m="1304830">first</span> <span m="1305130">game</span> <span m="1305370">because</span> <span m="1305530">it's</span> <span m="1305660">been</span> <span m="1305830">played.</span> <span m="1307270">So</span> <span m="1307420">how</span> <span m="1307660">could</span> <span m="1307780">there</span> <span m="1307880">be</span> <span m="1308020">any</span> <span m="1308210">probability</span> <span m="1308980">there?</span> <span m="1309270">It</span> <span m="1309410">happened.</span> </p>
<p><span m="1311240">Well,</span> <span m="1311530">so</span> <span m="1311630">what</span> <span m="1311740">the</span> <span m="1311830">meaning</span> <span m="1312230">is</span> <span m="1312530">is</span> <span m="1313150">over</span> <span m="1313410">all</span> <span m="1313700">the</span> <span m="1313820">times</span> <span m="1314460">where</span> <span m="1314640">the</span> <span m="1315360">series</span> <span m="1315740">was</span> <span m="1315900">played,</span> <span m="1316740">sort</span> <span m="1316910">of</span> <span m="1316990">what</span> <span m="1317200">fraction</span> <span m="1317710">of</span> <span m="1317830">the time</span> <span m="1318150">did</span> <span m="1318210">the</span> <span m="1318350">team</span> <span m="1318570">that</span> <span m="1318670">won</span> <span m="1318810">the</span> <span m="1318910">series</span> <span m="1319400">win</span> <span m="1319660">the</span> <span m="1319740">first</span> <span m="1320030">game</span> <span m="1320440">is</span> <span m="1320580">one</span> <span m="1320760">way</span> <span m="1320890">you</span> <span m="1320990">could</span> <span m="1321130">think</span> <span m="1321340">about</span> <span m="1321790">it.</span> <span m="1323330">Or,</span> <span m="1323650">maybe</span> <span m="1324300">you</span> <span m="1324420">just</span> <span m="1324690">don't</span> <span m="1324920">know.</span> <span m="1325120">The</span> <span m="1325250">game</span> <span m="1325510">was</span> <span m="1325660">played.</span> <span m="1326080">You</span> <span m="1326355">know you</span> <span m="1326630">won</span> <span m="1326770">the</span> <span m="1326860">series.</span> <span m="1327190">But</span> <span m="1327320">you don't</span> <span m="1327530">know</span> <span m="1327890">who</span> <span m="1328010">won</span> <span m="1328210">the</span> <span m="1328290">first</span> <span m="1328550">game.</span> <span m="1329300">And</span> <span m="1329430">so</span> <span m="1329510">you</span> <span m="1329560">could</span> <span m="1329700">think</span> <span m="1329850">of</span> <span m="1329890">a</span> <span m="1329970">probability</span> <span m="1330490">still</span> <span m="1330720">being</span> <span m="1331010">there.</span> </p>
<p><span m="1332540">Now</span> <span m="1332850">when</span> <span m="1332990">you</span> <span m="1333080">think</span> <span m="1333320">about</span> <span m="1333610">it,</span> <span m="1334630">it</span> <span m="1334750">gets</span> <span m="1334920">me</span> <span m="1335070">confused</span> <span m="1335530">still.</span> <span m="1337220">But</span> <span m="1337370">just</span> <span m="1337910">think</span> <span m="1338110">about</span> <span m="1338430">it like</span> <span m="1338600">the</span> <span m="1338670">math.</span> <span m="1339840">It's</span> <span m="1340000">the</span> <span m="1340080">same</span> <span m="1340350">formula.</span> <span m="1342200">OK.</span> <span m="1342480">It doesn't</span> <span m="1342730">matter</span> <span m="1343020">which</span> <span m="1343790">happened</span> <span m="1344220">first</span> <span m="1344570">in</span> <span m="1344690">time.</span> <span m="1345800">You</span> <span m="1345900">use</span> <span m="1346230">the</span> <span m="1346310">same</span> <span m="1346820">mathematics.</span> <span m="1347750">In fact,</span> <span m="1347960">they</span> <span m="1348030">give</span> <span m="1348230">a</span> <span m="1348280">special</span> <span m="1348710">name</span> <span m="1349060">these</span> <span m="1349720">kinds</span> <span m="1349960">of</span> <span m="1350060">things.</span> <span m="1351310">They're</span> <span m="1351460">called</span> <span m="1351890">a</span> <span m="1352140">postieri</span> <span m="1353630">conditional</span> <span m="1354080">probabilities.</span> </p>
<p></p>
<p><span m="1370410">It's</span> <span m="1370550">a</span> <span m="1370610">fancy</span> <span m="1371070">name</span> <span m="1371570">for</span> <span m="1371700">just</span> <span m="1371950">saying</span> <span m="1373961">that</span> <span m="1374430">things</span> <span m="1374690">are</span> <span m="1374830">out</span> <span m="1375060">of</span> <span m="1375170">order</span> <span m="1375490">in</span> <span m="1375590">time.</span> <span m="1377810">All</span> <span m="1377930">right?</span> <span m="1378210">So</span> <span m="1378450">it's</span> <span m="1379580">a</span> <span m="1379690">probability</span> <span m="1380380">of</span> <span m="1380460">B</span> <span m="1380850">given</span> <span m="1381110">A</span> <span m="1384920">where</span> <span m="1385250">B</span> <span m="1385970">precedes</span> <span m="1386800">A</span> <span m="1387485">in</span> <span m="1387760">time.</span> <span m="1394100">All</span> <span m="1394210">right?</span> <span m="1396150">So</span> <span m="1396300">it's</span> <span m="1396400">the</span> <span m="1396470">same</span> <span m="1397310">math.</span> <span m="1397760">It's</span> <span m="1397880">just</span> <span m="1399100">they're</span> <span m="1399800">out</span> <span m="1400000">of</span> <span m="1400120">order.</span> </p>
<p><span m="1401500">So</span> <span m="1401640">let's</span> <span m="1401870">figure</span> <span m="1402210">out</span> <span m="1402680">the</span> <span m="1402770">probability</span> <span m="1404910">that</span> <span m="1405160">you</span> <span m="1405310">won</span> <span m="1405700">the</span> <span m="1405790">first</span> <span m="1406150">game</span> <span m="1407440">given</span> <span m="1407700">that</span> <span m="1407810">you</span> <span m="1407940">want</span> <span m="1408210">the</span> <span m="1408290">series.</span> <span m="1410090">Let's</span> <span m="1410350">figure</span> <span m="1410630">it</span> <span m="1410690">out.</span> <span m="1412570">So</span> <span m="1413590">I</span> <span m="1413720">want</span> <span m="1414130">probability</span> <span m="1414720">of</span> <span m="1414800">B</span> <span m="1415130">given</span> <span m="1415420">A</span> <span m="1415900">now</span> <span m="1416590">for</span> <span m="1416700">this</span> <span m="1416870">example.</span> <span m="1418330">Well,</span> <span m="1418590">it's</span> <span m="1418760">just</span> <span m="1418960">the</span> <span m="1419050">probability</span> <span m="1419730">of</span> <span m="1419810">B</span> <span m="1420230">and</span> <span m="1420590">A</span> <span m="1422620">over</span> <span m="1422930">the</span> <span m="1423040">probability</span> <span m="1423610">of</span> <span m="1423680">A.</span> </p>
<p><span m="1427590">We</span> <span m="1427770">already</span> <span m="1428050">computed</span> <span m="1428580">the</span> <span m="1428660">probability</span> <span m="1429180">of A</span> <span m="1429280">and</span> <span m="1429720">B.</span> <span m="1429930">That's</span> <span m="1430130">1/3</span> <span m="1430610">plus</span> <span m="1430850">1/18.</span> <span m="1436380">what's</span> <span m="1436610">the</span> <span m="1436710">probability</span> <span m="1437310">of</span> <span m="1437390">A,</span> <span m="1437890">the</span> <span m="1438270">probability</span> <span m="1438760">of</span> <span m="1438870">winning</span> <span m="1439350">the</span> <span m="1439440">first</span> <span m="1439820">game?</span> <span m="1444860">1/2.</span> <span m="1445205">It's</span> <span m="1445470">those</span> <span m="1445720">three</span> <span m="1445890">sample</span> <span m="1446260">points</span> <span m="1446530">and</span> <span m="1446710">they</span> <span m="1446980">better</span> <span m="1447280">add</span> <span m="1447420">up</span> <span m="1447520">to</span> <span m="1447650">1/2</span> <span m="1448130">because</span> <span m="1448290">we</span> <span m="1448380">sort</span> <span m="1448560">of</span> <span m="1448660">said,</span> <span m="1449170">the</span> <span m="1449620">probability</span> <span m="1450110">of the first</span> <span m="1450350">game's</span> <span m="1450590">1/2.</span> <span m="1452350">So</span> <span m="1452490">that's</span> <span m="1452740">over</span> <span m="1452980">1/2,</span> <span m="1454060">which</span> <span m="1454280">is</span> <span m="1454510">9/18.</span> </p>
<p><span m="1457780">Well</span> <span m="1457970">this</span> <span m="1458140">was</span> <span m="1458310">7/18</span> <span m="1459190">over</span> <span m="1459470">9/18.</span> <span m="1460200">It's</span> <span m="1460340">7/9.</span> <span m="1462980">So</span> <span m="1463440">the</span> <span m="1463540">probability</span> <span m="1464010">of</span> <span m="1464080">winning</span> <span m="1464390">the</span> <span m="1464480">first</span> <span m="1464770">game</span> <span m="1465110">given</span> <span m="1465560">that you</span> <span m="1465900">won</span> <span m="1465990">series</span> <span m="1466550">is</span> <span m="1466670">7/9.</span> <span m="1469950">Anybody</span> <span m="1470400">notice</span> <span m="1470650">anything</span> <span m="1470880">unusual</span> <span m="1471240">about</span> <span m="1471430">that</span> <span m="1471610">answer</span> <span m="1471970">here?</span> </p>
<p><span m="1474218">It's</span> <span m="1474670">the</span> <span m="1474770">same</span> <span m="1475090">as</span> <span m="1475190">the</span> <span m="1475330">answer</span> <span m="1475660">over</span> <span m="1475810">there.</span> <span m="1477780">Is</span> <span m="1477900">that</span> <span m="1478070">a</span> <span m="1478120">theorem?</span> <span m="1480990">No.</span> <span m="1481470">The</span> <span m="1481560">probability</span> <span m="1482080">of</span> <span m="1482150">A</span> <span m="1482310">given</span> <span m="1482610">B</span> <span m="1482790">is</span> <span m="1482960">not</span> <span m="1483390">always</span> <span m="1483920">the</span> <span m="1484020">probability</span> <span m="1484450">of</span> <span m="1484520">B</span> <span m="1484740">given</span> <span m="1485010">A.</span> <span m="1485310">It</span> <span m="1485470">was</span> <span m="1485970">in</span> <span m="1486090">this</span> <span m="1486310">case.</span> <span m="1487230">It</span> <span m="1487250">is</span> <span m="1487410">not</span> <span m="1487700">always</span> <span m="1488070">true.</span> <span m="1489070">In</span> <span m="1489100">fact,</span> <span m="1489470">we</span> <span m="1489570">could</span> <span m="1489590">make</span> <span m="1489860">a</span> <span m="1490000">simple</span> <span m="1490410">example</span> <span m="1492310">to</span> <span m="1492390">see</span> <span m="1492610">why</span> <span m="1492780">that's</span> <span m="1493040">not</span> <span m="1493270">always</span> <span m="1493600">the</span> <span m="1493680">case.</span> </p>
<p></p>
<p><span m="1507360">All</span> <span m="1507420">right.</span> <span m="1507690">So</span> <span m="1508020">say</span> <span m="1508700">here's</span> <span m="1509020">your</span> <span m="1509150">sample</span> <span m="1509550">space.</span> <span m="1511150">And</span> <span m="1511710">say</span> <span m="1512010">that</span> <span m="1512260">this</span> <span m="1512490">is</span> <span m="1512630">B</span> <span m="1514600">and</span> <span m="1514850">this</span> <span m="1515630">is</span> <span m="1515850">A.</span> <span m="1517970">What's</span> <span m="1518200">the</span> <span m="1518300">probability</span> <span m="1518860">of</span> <span m="1519050">A</span> <span m="1519390">given</span> <span m="1519750">B</span> <span m="1520440">in</span> <span m="1520540">this</span> <span m="1520730">case?</span> <span m="1523670">1.</span> <span m="1524330">If</span> <span m="1524470">you're</span> <span m="1524700">in</span> <span m="1524810">B--</span> <span m="1526120">wait.</span> <span m="1526330">No.</span> <span m="1526480">It's</span> <span m="1526640">not</span> <span m="1526840">1.</span> <span m="1528850">What's</span> <span m="1529040">the</span> <span m="1529120">probability</span> <span m="1529530">of</span> <span m="1529590">A</span> <span m="1529730">given</span> <span m="1530020">B</span> <span m="1530230">If</span> <span m="1530330">I</span> <span m="1530410">got</span> <span m="1530660">some--</span> <span m="1532820">probably</span> <span m="1533150">less</span> <span m="1533490">than</span> <span m="1533650">1.</span> <span m="1533970">Might</span> <span m="1534190">be</span> <span m="1534360">I've</span> <span m="1534480">drawn</span> <span m="1534890">it as</span> <span m="1535020">1/3</span> <span m="1535090">third</span> <span m="1535400">if</span> <span m="1535490">it</span> <span m="1535560">was</span> <span m="1535690">uniform.</span> </p>
<p><span m="1537210">But</span> <span m="1537340">in</span> <span m="1537390">this</span> <span m="1537590">case,</span> <span m="1537950">the</span> <span m="1538050">probability</span> <span m="1538460">of</span> <span m="1538550">A</span> <span m="1539570">given</span> <span m="1539840">B</span> <span m="1540730">is</span> <span m="1540900">less</span> <span m="1541200">than</span> <span m="1541350">1.</span> <span m="1541580">What's</span> <span m="1541790">the</span> <span m="1541870">probability</span> <span m="1542340">of</span> <span m="1542420">B</span> <span m="1542650">given</span> <span m="1542950">A?</span> <span m="1547240">1,</span> <span m="1548080">because</span> <span m="1548290">if</span> <span m="1548460">I'm in</span> <span m="1548640">A</span> <span m="1549630">I'm</span> <span m="1549810">definitely</span> <span m="1550330">in</span> <span m="1550380">B.</span> <span m="1553660">All</span> <span m="1553740">right.</span> <span m="1553950">So</span> <span m="1554030">that's</span> <span m="1554260">an</span> <span m="1554340">example</span> <span m="1554880">where</span> <span m="1555000">they</span> <span m="1555180">would</span> <span m="1555340">be</span> <span m="1555480">different.</span> <span m="1555940">And</span> <span m="1556050">that's</span> <span m="1556250">the</span> <span m="1556370">generic</span> <span m="1556980">case</span> <span m="1557440">is</span> <span m="1557600">they're</span> <span m="1558055">different.</span> <span m="1559450">All</span> <span m="1559600">right?</span> </p>
<p><span m="1561840">When</span> <span m="1562220">are</span> <span m="1562480">they</span> <span m="1562680">equal</span> <span m="1562990">because</span> <span m="1563120">they</span> <span m="1563240">were</span> <span m="1563360">equal</span> <span m="1563680">in</span> <span m="1563740">this</span> <span m="1563940">case?</span> <span m="1564360">What</span> <span m="1564570">makes</span> <span m="1564850">them</span> <span m="1566240">equal?</span> <span m="1566640">Let's</span> <span m="1566930">see.</span> <span m="1567330">When</span> <span m="1567590">does</span> <span m="1568495">the</span> <span m="1568750">probability</span> <span m="1569530">of</span> <span m="1569620">A</span> <span m="1569970">given</span> <span m="1570240">B</span> <span m="1571900">equal a</span> <span m="1572360">probability</span> <span m="1573100">of</span> <span m="1573190">B</span> <span m="1573510">given</span> <span m="1573820">A?</span> </p>
<p><span m="1574170">Let's</span> <span m="1574480">see.</span> <span m="1576135">Well,</span> <span m="1576560">If</span> <span m="1576650">I</span> <span m="1576710">plug-in</span> <span m="1577110">the</span> <span m="1577210">formula,</span> <span m="1578900">this</span> <span m="1579150">equals</span> <span m="1580360">the</span> <span m="1580420">probability</span> <span m="1580870">of</span> <span m="1581040">A and</span> <span m="1581840">B</span> <span m="1582990">over the</span> <span m="1583350">probability</span> <span m="1583900">of</span> <span m="1583990">B.</span> <span m="1586410">That</span> <span m="1586620">equals</span> <span m="1587710">the</span> <span m="1587800">probability</span> <span m="1588520">of</span> <span m="1588940">B</span> <span m="1590010">and</span> <span m="1590430">A</span> <span m="1591820">over a</span> <span m="1592210">probability</span> <span m="1592830">of A.</span> <span m="1595000">So</span> <span m="1595150">when</span> <span m="1595360">are</span> <span m="1595410">those</span> <span m="1595760">equal?</span> <span m="1599580">Yeah.</span> <span m="1599930">When</span> <span m="1600030">probability</span> <span m="1600680">A</span> <span m="1601100">equals</span> <span m="1601390">probability</span> <span m="1601990">B.</span> <span m="1602915">All right.</span> <span m="1603380">So</span> <span m="1603510">that's</span> <span m="1604430">one</span> <span m="1604760">case.</span> </p>
<p><span m="1608152">What's</span> <span m="1608620">the</span> <span m="1608810">other</span> <span m="1608980">case?</span> <span m="1614230">Yeah--</span> <span m="1614560">when</span> <span m="1614720">it's</span> <span m="1614950">0.</span> <span m="1615520">Probability--</span> <span m="1616060">there's</span> <span m="1616220">no</span> <span m="1616390">intersection.</span> <span m="1617460">Probability</span> <span m="1617790">of</span> <span m="1617880">A</span> <span m="1618100">intersect</span> <span m="1618480">B</span> <span m="1618600">is</span> <span m="1618760">0.</span> <span m="1619730">That's</span> <span m="1619970">the</span> <span m="1620060">other</span> <span m="1620220">case.</span> <span m="1627400">All right.</span> <span m="1627570">But</span> <span m="1627760">usually</span> <span m="1629170">these</span> <span m="1629430">conditions</span> <span m="1629850">won't</span> <span m="1630030">apply--</span> <span m="1630400">just</span> <span m="1630630">happened</span> <span m="1631080">to</span> <span m="1631250">in</span> <span m="1631330">this</span> <span m="1631500">example</span> <span m="1633150">by</span> <span m="1633300">coincidence.</span> </p>
<p><span m="1636320">Any</span> <span m="1636470">questions</span> <span m="1639050">about</span> <span m="1639970">that?</span> <span m="1641750">All</span> <span m="1641970">right.</span> <span m="1643940">Yeah.</span> <span m="1644150">So the</span> <span m="1644270">math</span> <span m="1644560">is</span> <span m="1644670">the</span> <span m="1644770">same</span> <span m="1645260">with</span> <span m="1645470">a postieri</span> <span m="1645670">probabilities.</span> <span m="1646330">It's</span> <span m="1646570">really,</span> <span m="1647300">really</span> <span m="1647500">easy.</span> </p>
<p><span m="1649210">All</span> <span m="1649610">right.</span> <span m="1649830">So</span> <span m="1649950">let's</span> <span m="1650210">do</span> <span m="1650850">another</span> <span m="1651250">simple</span> <span m="1651750">example</span> <span m="1652250">that'll</span> <span m="1652480">start</span> <span m="1652910">to</span> <span m="1653450">maybe</span> <span m="1653700">be</span> <span m="1653850">a</span> <span m="1653900">little</span> <span m="1654060">more</span> <span m="1654290">confusing.</span> <span m="1658470">Say</span> <span m="1658700">we've</span> <span m="1658900">got</span> <span m="1660120">two</span> <span m="1660330">coins.</span> </p>
<p></p>
<p><span m="1673510">One</span> <span m="1673810">of</span> <span m="1673890">them</span> <span m="1674090">is</span> <span m="1674200">a</span> <span m="1674260">fair</span> <span m="1674600">coin.</span> <span m="1679200">And</span> <span m="1679450">by</span> <span m="1679570">that,</span> <span m="1679800">I</span> <span m="1679880">mean</span> <span m="1680180">the</span> <span m="1680290">probability</span> <span m="1681060">comes</span> <span m="1681360">up</span> <span m="1681530">heads</span> <span m="1682670">is</span> <span m="1682880">the</span> <span m="1682970">same</span> <span m="1683320">as</span> <span m="1683400">the</span> <span m="1683500">probability</span> <span m="1684060">comes</span> <span m="1684380">up</span> <span m="1684560">tails</span> <span m="1685600">is</span> <span m="1685770">1/2.</span> <span m="1686980">The</span> <span m="1687130">other</span> <span m="1687280">one</span> <span m="1687550">is</span> <span m="1687690">an</span> <span m="1687840">unfair</span> <span m="1688360">coin.</span> <span m="1692440">And</span> <span m="1692590">in</span> <span m="1692660">this</span> <span m="1692890">case,</span> <span m="1693310">that</span> <span m="1693530">means</span> <span m="1694080">it's</span> <span m="1694280">always</span> <span m="1694730">heads.</span> </p>
<p><span m="1696056">The</span> <span m="1696430">probability</span> <span m="1696820">of</span> <span m="1696890">heads</span> <span m="1697260">is</span> <span m="1697390">1.</span> <span m="1697930">The</span> <span m="1698310">probability</span> <span m="1698960">of</span> <span m="1699060">tails</span> <span m="1699920">is</span> <span m="1700470">0.</span> <span m="1701392">All right?</span> <span m="1702540">I've</span> <span m="1702720">got</span> <span m="1703200">two</span> <span m="1703410">such</span> <span m="1703730">coins</span> <span m="1704110">here.</span> </p>
<p><span m="1705370">All right.</span> <span m="1706330">Here</span> <span m="1706630">is</span> <span m="1706700">the</span> <span m="1706850">unfair</span> <span m="1707340">coin--</span> <span m="1708960">heads</span> <span m="1709230">and</span> <span m="1709320">heads.</span> <span m="1710870">Actually,</span> <span m="1711260">they</span> <span m="1711350">make</span> <span m="1711570">these</span> <span m="1711760">things</span> <span m="1711970">look</span> <span m="1712120">like</span> <span m="1712250">quarters</span> <span m="1712590">sometimes.</span> <span m="1713480">Here's</span> <span m="1713940">the</span> <span m="1714060">fair</span> <span m="1714350">coin--</span> <span m="1715580">heads</span> <span m="1715890">and</span> <span m="1716020">tails.</span> <span m="1718200">All right.</span> </p>
<p><span m="1718900">Now</span> <span m="1720320">suppose</span> <span m="1720840">I</span> <span m="1720950">pick</span> <span m="1721460">one</span> <span m="1721650">of</span> <span m="1721720">these</span> <span m="1722030">at</span> <span m="1722200">random,</span> <span m="1724500">50-50,</span> <span m="1725410">I</span> <span m="1725530">pick</span> <span m="1725790">one</span> <span m="1725920">of</span> <span m="1725970">these</span> <span m="1726170">things,</span> <span m="1729310">and</span> <span m="1730440">I</span> <span m="1730590">flip</span> <span m="1731000">it,</span> <span m="1732500">which</span> <span m="1732680">I'm</span> <span m="1732790">doing</span> <span m="1733000">behind</span> <span m="1733270">my</span> <span m="1733400">back,</span> <span m="1735040">and</span> <span m="1735290">lo</span> <span m="1735630">and</span> <span m="1735730">behold,</span> <span m="1737570">it</span> <span m="1737610">comes</span> <span m="1737910">out</span> <span m="1738090">and,</span> <span m="1738180">you</span> <span m="1738320">see</span> <span m="1738590">a heads.</span> <span m="1742240">What's</span> <span m="1742560">the</span> <span m="1742670">probability</span> <span m="1744090">I'm</span> <span m="1744290">holding</span> <span m="1745920">the</span> <span m="1746050">fair</span> <span m="1746380">coin?</span> <span m="1751010">I</span> <span m="1751150">picked</span> <span m="1751730">the</span> <span m="1751830">coin,</span> <span m="1752380">50-50,</span> <span m="1753250">behind</span> <span m="1753620">my</span> <span m="1753750">back.</span> <span m="1755700">So</span> <span m="1755960">one</span> <span m="1756180">answer</span> <span m="1756610">is,</span> <span m="1757260">I</span> <span m="1757400">picked</span> <span m="1757880">the</span> <span m="1757980">fair</span> <span m="1758240">coin</span> <span m="1758500">with</span> <span m="1759080">50%</span> <span m="1759600">probability.</span> </p>
<p><span m="1761420">But</span> <span m="1761640">then</span> <span m="1761880">I</span> <span m="1762000">flipped</span> <span m="1762490">it</span> <span m="1762590">behind</span> <span m="1762930">my</span> <span m="1763080">back</span> <span m="1764210">and</span> <span m="1764390">I</span> <span m="1764470">showed</span> <span m="1764880">you</span> <span m="1764970">the</span> <span m="1765080">result.</span> <span m="1766130">And</span> <span m="1766480">you</span> <span m="1766590">see</span> <span m="1767550">heads.</span> <span m="1768120">Of</span> <span m="1768620">course,</span> <span m="1768830">if I'd</span> <span m="1769020">have</span> <span m="1769060">shown</span> <span m="1769410">you tails,</span> <span m="1771250">You would have</span> <span m="1771460">known</span> <span m="1771760">for</span> <span m="1771890">sure</span> <span m="1772850">it</span> <span m="1772980">was</span> <span m="1773130">the</span> <span m="1773230">fair</span> <span m="1773480">coin</span> <span m="1773850">because</span> <span m="1774110">that's</span> <span m="1774370">the only</span> <span m="1774460">one with the</span> <span m="1774570">tails.</span> </p>
<p><span m="1774960">But</span> <span m="1775230">you</span> <span m="1775340">don't</span> <span m="1775560">know</span> <span m="1775710">for</span> <span m="1775860">sure</span> <span m="1776110">now.</span> <span m="1776400">You</span> <span m="1776540">see</span> <span m="1776770">a heads.</span> <span m="1778400">What's</span> <span m="1778650">the</span> <span m="1778730">probability</span> <span m="1779300">this</span> <span m="1779550">is</span> <span m="1779670">the</span> <span m="1779780">fair</span> <span m="1780040">coin</span> <span m="1780740">given</span> <span m="1781090">that</span> <span m="1781190">you</span> <span m="1781280">saw</span> <span m="1782240">a</span> <span m="1782290">heads</span> <span m="1782760">after</span> <span m="1782990">the</span> <span m="1783090">flip?</span> </p>
<p><span m="1785470">How</span> <span m="1785640">many</span> <span m="1785840">people</span> <span m="1786140">think</span> <span m="1786370">1/2?</span> <span m="1788370">After</span> <span m="1788690">all,</span> <span m="1788820">I picked</span> <span m="1789100">it</span> <span m="1789170">with</span> <span m="1789290">probability</span> <span m="1789760">1/2.</span> <span m="1790940">How</span> <span m="1791100">many</span> <span m="1791330">people</span> <span m="1791690">think</span> <span m="1791960">it's</span> <span m="1792140">less</span> <span m="1792520">than</span> <span m="1792680">1/2?</span> <span m="1794890">Good.</span> <span m="1795680">OK.</span> </p>
<p><span m="1797530">Somebody</span> <span m="1797920">even</span> <span m="1798120">said</span> <span m="1798820">1/3.</span> <span m="1800110">Does that</span> <span m="1800280">sound</span> <span m="1800520">right?</span> <span m="1802530">A</span> <span m="1802550">couple</span> <span m="1802820">people</span> <span m="1803090">like</span> <span m="1803280">1/3.</span> <span m="1804870">OK.</span> <span m="1807280">All</span> <span m="1807420">right.</span> </p>
<p><span m="1807660">Now,</span> <span m="1808610">part</span> <span m="1808860">of</span> <span m="1808930">what</span> <span m="1809110">makes</span> <span m="1809380">this</span> <span m="1809570">tricky</span> <span m="1809990">is</span> <span m="1810130">I</span> <span m="1810200">told</span> <span m="1810510">you</span> <span m="1810600">I</span> <span m="1810690">picked</span> <span m="1811190">the</span> <span m="1811290">coin</span> <span m="1811590">with</span> <span m="1811740">50%</span> <span m="1812800">probability.</span> <span m="1813950">But</span> <span m="1814170">then</span> <span m="1814340">I</span> <span m="1814400">gave</span> <span m="1814690">you</span> <span m="1814830">information.</span> <span m="1815730">So</span> <span m="1815890">I've</span> <span m="1816390">conditioned</span> <span m="1817670">the</span> <span m="1817760">problem.</span> </p>
<p><span m="1820040">And</span> <span m="1820090">so</span> <span m="1820200">this</span> <span m="1820370">is</span> <span m="1820420">one</span> <span m="1820580">of</span> <span m="1820620">those</span> <span m="1820800">things</span> <span m="1821010">you</span> <span m="1821190">could</span> <span m="1821280">have</span> <span m="1821430">an</span> <span m="1821510">ask</span> <span m="1821760">Marilyn</span> <span m="1822120">about.</span> <span m="1823130">Is</span> <span m="1823260">it</span> <span m="1823340">1/2</span> <span m="1823790">or</span> <span m="1823870">is</span> <span m="1823960">it</span> <span m="1824040">1/3?</span> <span m="1824350">Because</span> <span m="1824490">I</span> <span m="1824920">picked</span> <span m="1825140">it</span> <span m="1825210">with</span> <span m="1825310">50%</span> <span m="1825860">chance,</span> <span m="1826530">what</span> <span m="1826970">does</span> <span m="1827080">the</span> <span m="1827180">information</span> <span m="1828840">do</span> <span m="1829040">for</span> <span m="1829330">you?</span> <span m="1830900">Now,</span> <span m="1831020">I'll</span> <span m="1831100">give</span> <span m="1831250">you</span> <span m="1831370">a</span> <span m="1831410">clue.</span> </p>
<p><span m="1834190">Bobo</span> <span m="1834700">might</span> <span m="1834930">have</span> <span m="1835050">written</span> <span m="1835340">in</span> <span m="1835790">and</span> <span m="1836040">said</span> <span m="1836330">it's</span> <span m="1836490">1/2.</span> <span m="1837660">And</span> <span m="1837830">his</span> <span m="1838070">proof</span> <span m="1838560">is that</span> <span m="1838810">three</span> <span m="1839040">other</span> <span m="1839180">mathematicians</span> <span m="1839840">agreed</span> <span m="1840190">with</span> <span m="1840300">him.</span> </p>
<p><span m="1840450">[LAUGHTER]</span> </p>
<p><span m="1841640">All</span> <span m="1841810">right?</span> <span m="1843690">OK.</span> <span m="1844550">So</span> <span m="1845800">let's</span> <span m="1846040">figure</span> <span m="1846300">it</span> <span m="1846370">out.</span> <span m="1848030">And</span> <span m="1848200">really</span> <span m="1849160">it's</span> <span m="1849570">very</span> <span m="1849810">simple.</span> <span m="1850320">It's</span> <span m="1850410">just</span> <span m="1851560">drawing</span> <span m="1852030">out</span> <span m="1852420">the</span> <span m="1852520">tree</span> <span m="1853830">and</span> <span m="1854020">computing</span> <span m="1854550">the</span> <span m="1854630">conditional</span> <span m="1855040">probability.</span> </p>
<p><span m="1856480">So</span> <span m="1856590">we're</span> <span m="1856730">going</span> <span m="1856820">to</span> <span m="1856910">do</span> <span m="1857050">the</span> <span m="1857450">same</span> <span m="1857960">thing</span> <span m="1859200">over</span> <span m="1859550">and</span> <span m="1859620">over</span> <span m="1859840">again</span> <span m="1860070">because</span> <span m="1860230">it</span> <span m="1860340">just</span> <span m="1860560">works</span> <span m="1860820">for</span> <span m="1860940">every</span> <span m="1861140">problem.</span> <span m="1863464">Of</span> <span m="1863925">course,</span> <span m="1864470">you</span> <span m="1864560">could</span> <span m="1864660">imagine</span> <span m="1865570">debating</span> <span m="1866120">this</span> <span m="1867220">for</span> <span m="1867410">awhile,</span> <span m="1867950">arguing</span> <span m="1868320">with</span> <span m="1868450">somebody.</span> <span m="1868890">Is</span> <span m="1868980">it</span> <span m="1869040">1/2</span> <span m="1869300">or</span> <span m="1869450">1/3?</span> <span m="1870200">Much</span> <span m="1870420">simpler</span> <span m="1870790">just</span> <span m="1871060">to</span> <span m="1872260">do</span> <span m="1872480">it.</span> </p>
<p><span m="1874690">So</span> <span m="1874830">the</span> <span m="1874920">first</span> <span m="1875200">thing</span> <span m="1875350">is</span> <span m="1875450">we</span> <span m="1875550">have,</span> <span m="1875840">which</span> <span m="1875970">coin</span> <span m="1876300">is picked?</span> <span m="1879300">So</span> <span m="1879480">it</span> <span m="1879560">could</span> <span m="1879710">be</span> <span m="1879840">fair--</span> <span m="1881350">and</span> <span m="1881480">I</span> <span m="1881560">told</span> <span m="1881940">you</span> <span m="1882040">that</span> <span m="1882270">happens</span> <span m="1882630">with</span> <span m="1882710">probability</span> <span m="1883140">1/2--</span> <span m="1884120">or</span> <span m="1884450">unfair,</span> <span m="1886650">which</span> <span m="1886870">is</span> <span m="1887000">also</span> <span m="1887390">1/2.</span> <span m="1889300">Then</span> <span m="1889750">we</span> <span m="1889860">have</span> <span m="1890500">the</span> <span m="1890590">flip.</span> <span m="1893290">The</span> <span m="1893390">fair</span> <span m="1893710">coin</span> <span m="1894470">is</span> <span m="1894640">equally</span> <span m="1895020">likely</span> <span m="1895370">to</span> <span m="1895420">be</span> <span m="1895570">heads</span> <span m="1895870">or</span> <span m="1895960">tails,</span> <span m="1897260">each</span> <span m="1897450">with</span> <span m="1897580">1/2.</span> <span m="1899170">The</span> <span m="1899330">unfair</span> <span m="1899760">coin,</span> <span m="1901060">guaranteed</span> <span m="1901610">to</span> <span m="1901670">be</span> <span m="1901810">heads,</span> <span m="1902890">probability</span> <span m="1903440">1.</span> </p>
<p><span m="1906190">All</span> <span m="1906380">right.</span> <span m="1906730">Now</span> <span m="1906960">we</span> <span m="1907060">get</span> <span m="1907240">the</span> <span m="1908680">sample</span> <span m="1909050">point</span> <span m="1909300">outcomes.</span> <span m="1910510">It's</span> <span m="1910770">fair</span> <span m="1911290">in</span> <span m="1911410">heads</span> <span m="1911920">with</span> <span m="1912020">the</span> <span m="1912090">probability</span> <span m="1912620">1/4,</span> <span m="1914310">fair</span> <span m="1914920">in tails,</span> <span m="1915650">probability</span> <span m="1916110">1/4,</span> <span m="1917420">unfair</span> <span m="1917990">in</span> <span m="1918050">heads,</span> <span m="1918490">probability</span> <span m="1918920">1/2.</span> <span m="1921580">Now</span> <span m="1921830">we</span> <span m="1921930">define</span> <span m="1922350">the</span> <span m="1922470">events</span> <span m="1922800">of</span> <span m="1922900">interest.</span> <span m="1925070">A</span> <span m="1925400">is</span> <span m="1925540">going</span> <span m="1925780">to</span> <span m="1925890">be</span> <span m="1926180">that</span> <span m="1926280">we</span> <span m="1926400">chose</span> <span m="1926800">the</span> <span m="1926890">fair</span> <span m="1927120">coin.</span> <span m="1933670">And</span> <span m="1933840">B</span> <span m="1934880">is</span> <span m="1935310">at</span> <span m="1935400">the</span> <span m="1935490">result,</span> <span m="1935930">is</span> <span m="1936060">heads.</span> <span m="1939980">And</span> <span m="1940060">of</span> <span m="1940130">course</span> <span m="1940370">what</span> <span m="1940500">I</span> <span m="1940540">want</span> <span m="1940870">to</span> <span m="1940920">know</span> <span m="1942150">is</span> <span m="1942330">the</span> <span m="1942430">probability</span> <span m="1943720">that</span> <span m="1943820">I</span> <span m="1943930">chose</span> <span m="1944350">the</span> <span m="1944440">fair</span> <span m="1944710">coin</span> <span m="1945320">given</span> <span m="1945640">that</span> <span m="1945820">I</span> <span m="1945910">saw</span> <span m="1946280">a heads.</span> </p>
<p><span m="1949610">So</span> <span m="1949880">to</span> <span m="1949990">do</span> <span m="1950180">that</span> <span m="1950610">we</span> <span m="1950750">plug in</span> <span m="1951100">our</span> <span m="1951190">formula.</span> <span m="1952020">That's</span> <span m="1952330">just</span> <span m="1952680">the</span> <span m="1954000">probability</span> <span m="1954800">of</span> <span m="1955195">A</span> <span m="1955470">and</span> <span m="1956360">B</span> <span m="1957750">over</span> <span m="1958100">the</span> <span m="1958210">probability</span> <span m="1958660">of</span> <span m="1958740">B.</span> <span m="1961290">And</span> <span m="1961540">to</span> <span m="1961630">compute</span> <span m="1962190">that</span> <span m="1963290">I</span> <span m="1963380">got</span> <span m="1963550">to</span> <span m="1963610">figure</span> <span m="1963910">out</span> <span m="1964370">the</span> <span m="1964450">probability</span> <span m="1965100">of A</span> <span m="1965460">and</span> <span m="1965770">B</span> <span m="1965890">and</span> <span m="1965970">the</span> <span m="1966060">probability</span> <span m="1966470">of</span> <span m="1966530">B.</span> </p>
<p><span m="1967240">So</span> <span m="1967630">I'll</span> <span m="1967810">make</span> <span m="1968030">my</span> <span m="1968550">diagram.</span> <span m="1969110">A</span> <span m="1969400">here,</span> <span m="1970160">B</span> <span m="1970470">here,</span> <span m="1971750">A and</span> <span m="1972170">B.</span> <span m="1974090">A</span> <span m="1974360">is the</span> <span m="1974600">event</span> <span m="1974970">I</span> <span m="1975050">chose</span> <span m="1975370">the</span> <span m="1975460">fair</span> <span m="1975710">coin.</span> <span m="1977570">That's these</span> <span m="1977840">guys.</span> <span m="1980050">B</span> <span m="1980310">is</span> <span m="1980440">the</span> <span m="1980580">event</span> <span m="1980960">the</span> <span m="1981040">result</span> <span m="1981440">is</span> <span m="1981550">heads.</span> <span m="1982850">That's</span> <span m="1983030">this</span> <span m="1983270">one</span> <span m="1984730">and</span> <span m="1985150">this</span> <span m="1985330">one.</span> <span m="1988070">And</span> <span m="1988540">A</span> <span m="1988750">intersect</span> <span m="1989230">B,</span> <span m="1990270">That's</span> <span m="1990590">the</span> <span m="1990720">only</span> <span m="1990910">point.</span> <span m="1992730">So</span> <span m="1992860">this</span> <span m="1993060">is</span> <span m="1993260">really</span> <span m="1993630">easy</span> <span m="1993910">to</span> <span m="1993990">compute</span> <span m="1994400">now.</span> </p>
<p><span m="1996450">What's</span> <span m="1996650">the</span> <span m="1996750">probability</span> <span m="1997250">of</span> <span m="1997320">A and</span> <span m="1997740">B?</span> <span m="2000770">1/4.</span> <span m="2001490">It's</span> <span m="2001630">just</span> <span m="2001860">that</span> <span m="2002080">sample</span> <span m="2002410">point.</span> <span m="2004270">What's</span> <span m="2004520">the</span> <span m="2004610">probability</span> <span m="2005400">of</span> <span m="2005590">B?</span> <span m="2008400">3/4,</span> <span m="2009330">1/4</span> <span m="2009800">plus</span> <span m="2010070">1/2.</span> <span m="2013100">So</span> <span m="2013660">the</span> <span m="2013760">probability</span> <span m="2014320">of</span> <span m="2014440">A</span> <span m="2014680">given</span> <span m="2014980">B</span> <span m="2015380">is</span> <span m="2015570">1/3.</span> </p>
<p><span m="2018650">Really</span> <span m="2019160">simple</span> <span m="2019650">to</span> <span m="2019730">answer</span> <span m="2020100">this</span> <span m="2020350">question.</span> <span m="2021400">Just</span> <span m="2022210">don't</span> <span m="2022370">even</span> <span m="2022590">think</span> <span m="2022880">about</span> <span m="2023160">it.</span> <span m="2023300">Just</span> <span m="2023830">write</span> <span m="2024060">down</span> <span m="2024240">the</span> <span m="2024330">tree</span> <span m="2025230">when</span> <span m="2025350">you</span> <span m="2025430">get</span> <span m="2025610">these</span> <span m="2025820">things.</span> <span m="2026460">So</span> <span m="2026910">much</span> <span m="2027270">easier</span> <span m="2027670">just</span> <span m="2027930">to</span> <span m="2028030">write</span> <span m="2028260">the</span> <span m="2028330">tree</span> <span m="2028540">down.</span> </p>
<p><span m="2030752">All right.</span> <span m="2031720">Now</span> <span m="2034060">the</span> <span m="2034170">key</span> <span m="2034460">here</span> <span m="2034750">is</span> <span m="2034840">we</span> <span m="2034970">knew</span> <span m="2035150">the</span> <span m="2035260">probability</span> <span m="2037690">of</span> <span m="2038120">picking</span> <span m="2038480">the</span> <span m="2038560">fair</span> <span m="2038850">coin</span> <span m="2039090">in</span> <span m="2039160">the</span> <span m="2039240">first</span> <span m="2039570">place.</span> <span m="2041445">Maybe</span> <span m="2041950">it's</span> <span m="2042170">worth</span> <span m="2042460">writing</span> <span m="2042730">down</span> <span m="2042940">what</span> <span m="2043090">happens</span> <span m="2043610">if</span> <span m="2044140">that's</span> <span m="2044370">a</span> <span m="2044410">variable--</span> <span m="2045500">sum</span> <span m="2045730">variable</span> <span m="2046140">P.</span> <span m="2046420">Let's</span> <span m="2046640">do</span> <span m="2046770">that.</span> </p>
<p><span m="2055239">For</span> <span m="2055330">example,</span> <span m="2057000">what</span> <span m="2057300">if</span> <span m="2057420">I</span> <span m="2057500">hadn't</span> <span m="2057850">told</span> <span m="2058260">you</span> <span m="2058429">the</span> <span m="2058550">probability</span> <span m="2060130">that</span> <span m="2060260">I</span> <span m="2060350">picked</span> <span m="2060690">the</span> <span m="2060750">fair</span> <span m="2061020">coin?</span> <span m="2062690">I</span> <span m="2062870">just</span> <span m="2063420">picked</span> <span m="2063699">one</span> <span m="2064199">and</span> <span m="2064380">flipped</span> <span m="2064679">it.</span> <span m="2066420">Think</span> <span m="2066550">that'll</span> <span m="2066750">change</span> <span m="2067070">the</span> <span m="2067199">answer?</span> <span m="2070130">It</span> <span m="2070630">should</span> <span m="2071449">because</span> <span m="2071719">you</span> <span m="2071830">got</span> <span m="2071969">to</span> <span m="2072020">plug</span> <span m="2072250">something</span> <span m="2072670">in</span> <span m="2072760">there</span> <span m="2072940">for</span> <span m="2073040">the</span> <span m="2073400">1/2</span> <span m="2073920">for</span> <span m="2073989">this</span> <span m="2074150">to</span> <span m="2074260">work.</span> </p>
<p><span m="2075139">So</span> <span m="2075170">let's</span> <span m="2075340">see</span> <span m="2075449">what</span> <span m="2075600">happens.</span> <span m="2076010">Say</span> <span m="2076230">I</span> <span m="2077590">picked</span> <span m="2077889">the</span> <span m="2078000">fair</span> <span m="2078300">coin</span> <span m="2080179">with</span> <span m="2080350">probability</span> <span m="2080889">P</span> <span m="2082239">and</span> <span m="2082400">the</span> <span m="2082540">unfair</span> <span m="2083070">coin</span> <span m="2084469">with</span> <span m="2084620">1</span> <span m="2084870">minus</span> <span m="2085270">P.</span> <span m="2086880">And</span> <span m="2087070">this</span> <span m="2087239">is</span> <span m="2087340">the</span> <span m="2087440">same</span> <span m="2087850">heads</span> <span m="2088270">and</span> <span m="2088409">tails,</span> <span m="2089860">1/2,</span> <span m="2091070">1/2.</span> <span m="2092429">Heads,</span> <span m="2093080">the</span> <span m="2093190">probability</span> <span m="2093650">1.</span> <span m="2096239">Well</span> <span m="2096389">now,</span> <span m="2096710">instead</span> <span m="2096969">of</span> <span m="2097110">1/4</span> <span m="2097550">I</span> <span m="2097610">get</span> <span m="2097790">P</span> <span m="2098010">over</span> <span m="2098200">2</span> <span m="2100350">up</span> <span m="2100520">here.</span> <span m="2101520">And</span> <span m="2101720">this</span> <span m="2101930">is</span> <span m="2102050">now</span> <span m="2102330">1</span> <span m="2102610">minus</span> <span m="2103010">P</span> <span m="2103230">instead</span> <span m="2103520">of</span> <span m="2103650">1/2.</span> </p>
<p><span m="2106580">So</span> <span m="2106750">the</span> <span m="2106860">probability</span> <span m="2108130">of</span> <span m="2108320">A</span> <span m="2108500">given</span> <span m="2108800">B</span> <span m="2111070">is</span> <span m="2111360">the</span> <span m="2111440">probability</span> <span m="2111890">of</span> <span m="2112060">A</span> <span m="2112340">and B</span> <span m="2112736">is</span> <span m="2113132">p</span> <span m="2113530">over</span> <span m="2113710">2.</span> <span m="2116220">And</span> <span m="2116420">the</span> <span m="2116500">probability</span> <span m="2117020">of</span> <span m="2117090">B</span> <span m="2118960">is</span> <span m="2119180">P</span> <span m="2119470">over</span> <span m="2119650">2</span> <span m="2119890">plus</span> <span m="2120200">1</span> <span m="2120340">minus</span> <span m="2120660">P.</span> <span m="2123670">That's</span> <span m="2123970">P</span> <span m="2124220">over</span> <span m="2124450">2</span> <span m="2124780">up</span> <span m="2124870">top,</span> <span m="2126580">one</span> <span m="2127470">minus</span> <span m="2127980">P</span> <span m="2128190">over</span> <span m="2128350">2,</span> <span m="2131100">and</span> <span m="2131500">that</span> <span m="2131710">is</span> <span m="2131940">all</span> <span m="2132730">multiplied</span> <span m="2133080">by--</span> <span m="2133220">what</span> <span m="2133585">am I</span> <span m="2133950">going</span> <span m="2134220">to</span> <span m="2134260">multiply--</span> <span m="2135200">2</span> <span m="2135520">here.</span> <span m="2135720">I'll</span> <span m="2135820">get</span> <span m="2136680">P</span> <span m="2137230">over</span> <span m="2137650">2</span> <span m="2137900">minus</span> <span m="2138250">P.</span> </p>
<p><span m="2141210">So</span> <span m="2141330">the</span> <span m="2141450">probability</span> <span m="2142090">with</span> <span m="2142200">which</span> <span m="2142610">I</span> <span m="2142810">picked</span> <span m="2143240">the</span> <span m="2143360">coin</span> <span m="2143660">to</span> <span m="2143730">start</span> <span m="2144080">with</span> <span m="2144810">impacts</span> <span m="2145570">the</span> <span m="2145750">answer</span> <span m="2146120">here.</span> <span m="2148060">For</span> <span m="2148280">example,</span> <span m="2149300">what</span> <span m="2149880">if</span> <span m="2151230">I</span> <span m="2151380">picked</span> <span m="2151790">the</span> <span m="2151990">unfair</span> <span m="2152640">coin</span> <span m="2153130">for</span> <span m="2153300">sure?</span> <span m="2155680">That</span> <span m="2155890">would</span> <span m="2156030">be</span> <span m="2156430">P</span> <span m="2156680">being</span> <span m="2157010">0.</span> <span m="2160820">Well,</span> <span m="2160920">the</span> <span m="2161040">probability</span> <span m="2161690">that</span> <span m="2161820">I</span> <span m="2161920">picked</span> <span m="2162230">the</span> <span m="2162290">fair</span> <span m="2162540">coin</span> <span m="2163130">is</span> <span m="2163450">0</span> <span m="2164390">over</span> <span m="2164640">2,</span> <span m="2164930">which</span> <span m="2165090">is</span> <span m="2165220">0.</span> <span m="2167340">All right</span> <span m="2167520">though--</span> <span m="2167730">even</span> <span m="2167910">know</span> <span m="2168010">I</span> <span m="2168110">showed</span> <span m="2168350">you</span> <span m="2168480">the</span> <span m="2168570">heads,</span> <span m="2168870">there's</span> <span m="2169070">no</span> <span m="2169340">chance</span> <span m="2169830">it</span> <span m="2169890">was</span> <span m="2170040">the</span> <span m="2170110">fair</span> <span m="2170340">coin</span> <span m="2170680">because</span> <span m="2171820">I</span> <span m="2171910">picked</span> <span m="2172260">the</span> <span m="2172410">unfair</span> <span m="2172810">coin</span> <span m="2173080">for</span> <span m="2173210">sure.</span> </p>
<p><span m="2179840">Same</span> <span m="2180100">thing</span> <span m="2180280">if</span> <span m="2180390">I</span> <span m="2180460">picked</span> <span m="2180760">the</span> <span m="2180800">fair</span> <span m="2181070">coin</span> <span m="2181350">for</span> <span m="2181480">sure,</span> <span m="2183020">better</span> <span m="2183230">be</span> <span m="2183340">the</span> <span m="2183450">case</span> <span m="2183840">this</span> <span m="2183900">is</span> <span m="2184010">1.</span> <span m="2185110">So</span> <span m="2185220">I</span> <span m="2185260">get</span> <span m="2185410">1</span> <span m="2185750">over</span> <span m="2185920">2</span> <span m="2186080">minus</span> <span m="2186420">1.</span> <span m="2186720">It's</span> <span m="2186910">1.</span> <span m="2190380">Any</span> <span m="2190510">questions?</span> <span m="2192610">So</span> <span m="2192820">it's</span> <span m="2193520">important</span> <span m="2194030">you</span> <span m="2194120">know</span> <span m="2194660">the</span> <span m="2194750">probability</span> <span m="2195360">I</span> <span m="2195440">picked</span> <span m="2196210">the</span> <span m="2196300">fair</span> <span m="2196500">coin</span> <span m="2196790">to start</span> <span m="2197030">with.</span> <span m="2197210">Otherwise,</span> <span m="2198150">you</span> <span m="2198730">can't</span> <span m="2198980">go</span> <span m="2199180">anywhere.</span> </p>
<p><span m="2202700">All</span> <span m="2202800">right.</span> <span m="2203930">What</span> <span m="2204220">if</span> <span m="2204400">I</span> <span m="2204470">do</span> <span m="2204580">the</span> <span m="2204690">same</span> <span m="2205140">game?</span> <span m="2206640">Pick</span> <span m="2206850">a</span> <span m="2206930">coin</span> <span m="2208160">with</span> <span m="2208610">probability</span> <span m="2209120">p.</span> <span m="2210910">But</span> <span m="2211080">now</span> <span m="2211290">I</span> <span m="2211420">flip</span> <span m="2211850">it</span> <span m="2212030">K</span> <span m="2212270">times.</span> <span m="2214291">Say</span> <span m="2214730">I</span> <span m="2214760">flip it</span> <span m="2214970">100</span> <span m="2215340">times.</span> <span m="2216400">And</span> <span m="2216600">every</span> <span m="2216940">time</span> <span m="2217280">it</span> <span m="2217380">comes</span> <span m="2217690">up</span> <span m="2217840">heads.</span> <span m="2221420">I</span> <span m="2221470">mean</span> <span m="2221570">you're</span> <span m="2221660">pretty</span> <span m="2221960">sure</span> <span m="2223400">you</span> <span m="2223510">got</span> <span m="2223720">the</span> <span m="2223860">unfair</span> <span m="2224210">coin</span> <span m="2224510">because</span> <span m="2224640">you</span> <span m="2224720">never</span> <span m="2225050">saw a</span> <span m="2225290">tails.</span> <span m="2225850">Right?</span> </p>
<p><span m="2227480">So</span> <span m="2227630">let's</span> <span m="2227830">do</span> <span m="2227990">that.</span> <span m="2228220">Let's</span> <span m="2228410">compute</span> <span m="2228890">that</span> <span m="2229240">scenario.</span> <span m="2231060">So</span> <span m="2231420">instead</span> <span m="2231720">of</span> <span m="2231790">a</span> <span m="2231850">single</span> <span m="2232240">heads</span> <span m="2232670">I</span> <span m="2232750">get</span> <span m="2234680">K</span> <span m="2235050">straight</span> <span m="2235440">heads</span> <span m="2236430">and</span> <span m="2236690">no</span> <span m="2237070">tails.</span> <span m="2239310">This</span> <span m="2239490">would</span> <span m="2239590">happen</span> <span m="2239920">with</span> <span m="2240070">1</span> <span m="2240270">over</span> <span m="2240440">2</span> <span m="2240610">to</span> <span m="2240680">the</span> <span m="2240800">K.</span> <span m="2242290">This</span> <span m="2242480">would</span> <span m="2242600">happen</span> <span m="2242930">with</span> <span m="2243060">1</span> <span m="2243350">minus</span> <span m="2243790">1</span> <span m="2244000">over</span> <span m="2244190">2</span> <span m="2244420">to the</span> <span m="2244550">K.</span> <span m="2247270">So</span> <span m="2247410">this</span> <span m="2247630">is</span> <span m="2247740">now</span> <span m="2248020">p</span> <span m="2248300">over</span> <span m="2248500">2</span> <span m="2248680">to</span> <span m="2248760">the</span> <span m="2248880">K.</span> <span m="2249340">This</span> <span m="2249540">is</span> <span m="2249640">now</span> <span m="2249910">P1</span> <span m="2251170">minus</span> <span m="2251790">2</span> <span m="2251980">to</span> <span m="2252030">the</span> <span m="2252130">minus</span> <span m="2252510">K.</span> </p>
<p><span m="2254870">Let's</span> <span m="2255080">recompute</span> <span m="2255590">the</span> <span m="2255670">probabilities.</span> <span m="2256380">I'm</span> <span m="2256690">going</span> <span m="2256950">somewhere</span> <span m="2257300">where</span> <span m="2257400">this.</span> <span m="2258910">Wait a minute.</span> </p>
<p></p>
<p><span m="2270340">So</span> <span m="2270460">now</span> <span m="2270750">we're</span> <span m="2270840">looking</span> <span m="2271120">at</span> <span m="2271200">the</span> <span m="2271320">event</span> <span m="2271800">that</span> <span m="2272950">B</span> <span m="2274150">is</span> <span m="2274450">K</span> <span m="2275030">straight</span> <span m="2275950">heads.</span> <span m="2279080">Come</span> <span m="2279360">up.</span> <span m="2282150">And</span> <span m="2282310">I</span> <span m="2282340">want</span> <span m="2282490">to</span> <span m="2282630">know</span> <span m="2282700">the</span> <span m="2282820">probability</span> <span m="2284240">that</span> <span m="2284390">I</span> <span m="2284470">picked</span> <span m="2284850">the</span> <span m="2284900">fair</span> <span m="2285180">coin</span> <span m="2285690">given</span> <span m="2286050">that</span> <span m="2286640">it</span> <span m="2286760">just</span> <span m="2286960">never</span> <span m="2287160">comes</span> <span m="2287420">up</span> <span m="2287540">tails.</span> <span m="2289730">The</span> <span m="2289810">math</span> <span m="2290090">is</span> <span m="2290200">the</span> <span m="2290290">same.</span> </p>
<p><span m="2297390">The</span> <span m="2297530">probability</span> <span m="2298220">now</span> <span m="2298540">that</span> <span m="2299030">I</span> <span m="2299120">picked</span> <span m="2299420">the</span> <span m="2299480">fair</span> <span m="2299720">coin and</span> <span m="2300070">got</span> <span m="2300260">k</span> <span m="2300490">straight</span> <span m="2300830">heads</span> <span m="2301220">is</span> <span m="2301360">just</span> <span m="2302560">p</span> <span m="2302850">times</span> <span m="2303140">2</span> <span m="2303310">to</span> <span m="2303360">the</span> <span m="2303440">minus</span> <span m="2303810">K.</span> <span m="2305490">The</span> <span m="2305600">probability</span> <span m="2306270">that</span> <span m="2306440">I</span> <span m="2306810">got</span> <span m="2307070">K</span> <span m="2307300">straight</span> <span m="2307660">heads</span> <span m="2309030">is</span> <span m="2311490">P</span> <span m="2311790">times</span> <span m="2312130">2</span> <span m="2312230">to</span> <span m="2312270">the</span> <span m="2312410">minus</span> <span m="2312790">K</span> <span m="2313450">plus</span> <span m="2313830">the</span> <span m="2313920">chance</span> <span m="2314210">I</span> <span m="2314260">picked</span> <span m="2314500">the</span> <span m="2314600">unfair</span> <span m="2314920">coin,</span> <span m="2315960">which</span> <span m="2316040">is</span> <span m="2316130">1</span> <span m="2316340">minus</span> <span m="2316710">P.</span> <span m="2318230">And</span> <span m="2318390">if</span> <span m="2318480">I</span> <span m="2318540">multiply</span> <span m="2318940">top</span> <span m="2319210">and</span> <span m="2319320">bottom</span> <span m="2319600">by</span> <span m="2319710">2</span> <span m="2319920">to the</span> <span m="2320060">K,</span> <span m="2320340">I</span> <span m="2320380">get</span> <span m="2320620">P</span> <span m="2321020">over</span> <span m="2322030">P</span> <span m="2322410">plus</span> <span m="2323000">to</span> <span m="2323240">the</span> <span m="2323380">K</span> <span m="2324240">1</span> <span m="2324580">minus</span> <span m="2324900">B.</span> </p>
<p><span m="2327612">All right.</span> <span m="2328070">So</span> <span m="2328240">it gets</span> <span m="2330050">very</span> <span m="2330840">unlikely</span> <span m="2331480">that</span> <span m="2331600">I've</span> <span m="2331710">got</span> <span m="2331890">the</span> <span m="2331970">fair</span> <span m="2332220">coin</span> <span m="2333050">here</span> <span m="2333290">as</span> <span m="2333410">K</span> <span m="2333610">gets</span> <span m="2333820">big.</span> <span m="2335280">Like</span> <span m="2335420">if</span> <span m="2335530">K</span> <span m="2335740">is</span> <span m="2335890">100</span> <span m="2337150">I</span> <span m="2337240">got</span> <span m="2337390">a</span> <span m="2337440">big</span> <span m="2337850">number</span> <span m="2338120">down</span> <span m="2338390">here.</span> <span m="2338660">And</span> <span m="2339090">basically</span> <span m="2339490">it's</span> <span m="2339710">0</span> <span m="2340040">chance--</span> <span m="2340460">close</span> <span m="2340680">to</span> <span m="2340750">0</span> <span m="2341000">chance</span> <span m="2341430">of</span> <span m="2343070">the</span> <span m="2343150">fair</span> <span m="2343420">coin.</span> </p>
<p><span m="2346910">But</span> <span m="2347090">now</span> <span m="2347340">say</span> <span m="2347530">I</span> <span m="2347600">do</span> <span m="2347710">the</span> <span m="2347820">following</span> <span m="2348300">experiment.</span> <span m="2350760">I</span> <span m="2350900">don't</span> <span m="2351130">tell</span> <span m="2351300">you</span> <span m="2351470">P.</span> <span m="2354330">But</span> <span m="2354480">I</span> <span m="2354570">pull</span> <span m="2354860">a</span> <span m="2354900">coin</span> <span m="2355260">out</span> <span m="2356820">and</span> <span m="2356980">100</span> <span m="2357530">flips</span> <span m="2357940">in</span> <span m="2358040">a</span> <span m="2358120">row</span> <span m="2359070">it's</span> <span m="2359300">heads.</span> <span m="2361380">Which</span> <span m="2361580">coin</span> <span m="2361840">do you</span> <span m="2361970">think</span> <span m="2362180">I</span> <span m="2362270">have?</span> <span m="2366340">I</span> <span m="2366460">flipped it</span> <span m="2366860">100</span> <span m="2367390">straight</span> <span m="2367730">times</span> <span m="2368100">and it's</span> <span m="2368310">heads</span> <span m="2368660">every</span> <span m="2368860">time.</span> </p>
<p><span m="2372550">Yeah.</span> <span m="2373310">There's</span> <span m="2373470">not</span> <span m="2373590">enough</span> <span m="2373760">information.</span> <span m="2374200">You</span> <span m="2374270">don't</span> <span m="2374440">know.</span> <span m="2374990">What</span> <span m="2375260">do</span> <span m="2375300">you</span> <span m="2375410">want</span> <span m="2375720">to</span> <span m="2375780">say?</span> <span m="2377840">You</span> <span m="2377990">want</span> <span m="2378240">to</span> <span m="2378300">say</span> <span m="2378510">it's</span> <span m="2378660">the</span> <span m="2378790">unfair</span> <span m="2379170">coin</span> <span m="2380890">but</span> <span m="2381050">you</span> <span m="2381170">have</span> <span m="2381370">no</span> <span m="2381570">idea</span> <span m="2382540">because</span> <span m="2382840">I</span> <span m="2382960">might</span> <span m="2383350">have</span> <span m="2383740">picked</span> <span m="2385550">the</span> <span m="2385670">fair</span> <span m="2385950">coin</span> <span m="2386250">with</span> <span m="2386350">probability</span> <span m="2386850">1,</span> <span m="2387190">in</span> <span m="2387260">which</span> <span m="2387440">case</span> <span m="2387650">it</span> <span m="2387770">is</span> <span m="2388000">the</span> <span m="2388100">fair</span> <span m="2388340">coin</span> <span m="2388790">and</span> <span m="2388870">it</span> <span m="2388940">just</span> <span m="2389070">was</span> <span m="2389200">unlucky</span> <span m="2389600">that it</span> <span m="2389750">came</span> <span m="2390020">up</span> <span m="2390650">heads</span> <span m="2391000">100</span> <span m="2391310">times</span> <span m="2391570">in</span> <span m="2391640">a</span> <span m="2391710">row.</span> <span m="2393270">But</span> <span m="2393420">it</span> <span m="2393490">could</span> <span m="2393710">be.</span> </p>
<p><span m="2394530">So</span> <span m="2394650">you</span> <span m="2394730">could</span> <span m="2394890">say</span> <span m="2395070">nothing</span> <span m="2396060">if</span> <span m="2396170">you</span> <span m="2396280">don't</span> <span m="2396540">know</span> <span m="2396880">the</span> <span m="2396990">probability</span> <span m="2397650">P.</span> <span m="2398730">Because</span> <span m="2399000">sure</span> <span m="2399290">enough,</span> <span m="2399780">if</span> <span m="2399830">I</span> <span m="2399940">plug in</span> <span m="2402180">P</span> <span m="2402445">being</span> <span m="2402710">1</span> <span m="2403060">here,</span> <span m="2405370">that</span> <span m="2405530">wipes</span> <span m="2405850">out the</span> <span m="2406010">2</span> <span m="2406280">to the</span> <span m="2406390">K</span> <span m="2406720">and I</span> <span m="2406790">just</span> <span m="2407010">get</span> <span m="2407180">probability</span> <span m="2407600">1.</span> <span m="2410600">OK?</span> </p>
<p><span m="2413120">All</span> <span m="2413220">right.</span> <span m="2413440">Now</span> <span m="2413920">when</span> <span m="2414050">this</span> <span m="2414240">comes</span> <span m="2414520">up</span> <span m="2416400">in</span> <span m="2416480">practice</span> <span m="2418510">is</span> <span m="2418920">with</span> <span m="2419260">things</span> <span m="2419470">like</span> <span m="2419700">polling.</span> <span m="2420560">Like,</span> <span m="2420690">we</span> <span m="2420770">just</span> <span m="2420960">had</span> <span m="2421120">an</span> <span m="2421180">election.</span> <span m="2422680">And</span> <span m="2422930">people</span> <span m="2423150">do</span> <span m="2423280">poles</span> <span m="2423690">ahead</span> <span m="2423900">of</span> <span m="2423980">time.</span> <span m="2425430">And</span> <span m="2426010">they</span> <span m="2426140">sample</span> <span m="2427000">thousands</span> <span m="2427630">of</span> <span m="2427730">voters</span> <span m="2428850">from</span> <span m="2429020">1%</span> <span m="2429610">of</span> <span m="2429650">the</span> <span m="2429730">population.</span> </p>
<p><span m="2431160">And</span> <span m="2431350">they</span> <span m="2431480">say,</span> <span m="2431850">OK,</span> <span m="2432180">that</span> <span m="2432390">60%</span> <span m="2433170">of</span> <span m="2433230">the</span> <span m="2433330">people</span> <span m="2433590">are</span> <span m="2433700">going</span> <span m="2433820">to</span> <span m="2433880">vote</span> <span m="2434620">Republican.</span> <span m="2437240">And</span> <span m="2437380">they</span> <span m="2437430">might</span> <span m="2437650">have</span> <span m="2437770">a</span> <span m="2437810">margin</span> <span m="2438210">of</span> <span m="2438300">error,</span> <span m="2438760">three</span> <span m="2438960">points,</span> <span m="2439750">whatever</span> <span m="2439960">that</span> <span m="2440150">means.</span> <span m="2440380">And we'll</span> <span m="2440560">figure</span> <span m="2440840">that</span> <span m="2441020">out</span> <span m="2441170">next</span> <span m="2441430">week.</span> <span m="2442580">What</span> <span m="2442810">does</span> <span m="2442960">that</span> <span m="2443220">tell</span> <span m="2443460">you</span> <span m="2443610">about</span> <span m="2443830">the</span> <span m="2444550">electorate</span> <span m="2445320">as</span> <span m="2445430">a</span> <span m="2445490">whole--</span> <span m="2445920">the</span> <span m="2446000">population</span> <span m="2447710">if</span> <span m="2447830">they</span> <span m="2448150">sample</span> <span m="2448740">1%</span> <span m="2449250">at</span> <span m="2449330">random,</span> <span m="2450490">60%</span> <span m="2451220">are</span> <span m="2451270">Republican.</span> <span m="2452180">Yeah?</span> </p>
<p><span m="2454572">AUDIENCE: [INAUDIBLE]</span> <span m="2457012">The</span> <span m="2457500">options</span> <span m="2457988">you have,</span> <span m="2458476">is it all</span> <span m="2458964">heads or is</span> <span m="2459452">it all tails?</span> <span m="2460428">It should</span> <span m="2460916">be one</span> <span m="2461404">option  all heads</span> <span m="2461892">and another</span> <span m="2462380">option at least</span> <span m="2462868">one tails.</span> </p>
<p><span m="2463860">PROFESSOR: You're</span> <span m="2464130">right.</span> <span m="2464910">Oops.</span> <span m="2466320">All right.</span> <span m="2466940">At least</span> <span m="2467195">one tail</span> <span m="2467450">for this one.</span> <span m="2467903">Yeah.</span> <span m="2468356">Good.</span> <span m="2470168">That is</span> <span m="2470621">true.</span> <span m="2471530">OK.</span> <span m="2473870">Any</span> <span m="2474030">questions</span> <span m="2474410">about</span> <span m="2474630">that</span> <span m="2475910">example?</span> </p>
<p><span m="2478550">OK.</span> <span m="2479610">Now</span> <span m="2479800">we're</span> <span m="2479900">back</span> <span m="2480130">to</span> <span m="2480230">the</span> <span m="2480340">election</span> <span m="2480990">and</span> <span m="2481150">there's</span> <span m="2481300">a</span> <span m="2481360">pole</span> <span m="2481740">that</span> <span m="2481850">says</span> <span m="2482880">they</span> <span m="2483000">sampled</span> <span m="2483480">1%</span> <span m="2483950">of</span> <span m="2483990">the</span> <span m="2484070">population</span> <span m="2484680">at</span> <span m="2484780">random</span> <span m="2486520">and</span> <span m="2486770">60%</span> <span m="2487610">said</span> <span m="2487770">they're</span> <span m="2487880">going</span> <span m="2488000">to</span> <span m="2488050">vote</span> <span m="2488250">Republican.</span> <span m="2489540">And</span> <span m="2489730">the</span> <span m="2489800">margin</span> <span m="2490200">of error is</span> <span m="2490650">3%</span> <span m="2491140">or</span> <span m="2491230">something.</span> <span m="2492570">What</span> <span m="2492710">does</span> <span m="2492820">that</span> <span m="2493000">tell</span> <span m="2493150">you</span> <span m="2493270">about</span> <span m="2493490">the</span> <span m="2493560">population</span> <span m="2493880">of</span> <span m="2494200">the</span> <span m="2494290">country?</span> </p>
<p><span m="2496510">Nothing.</span> <span m="2497900">That's</span> <span m="2498080">right.</span> <span m="2499560">It</span> <span m="2499750">is</span> <span m="2500140">what</span> <span m="2500330">it</span> <span m="2500420">is.</span> <span m="2502490">All</span> <span m="2502880">you</span> <span m="2503040">can</span> <span m="2503190">conclude</span> <span m="2504610">is</span> <span m="2504830">that</span> <span m="2505530">either</span> <span m="2506480">the</span> <span m="2506590">population</span> <span m="2507200">is</span> <span m="2507320">close</span> <span m="2507650">to</span> <span m="2507750">60%</span> <span m="2508360">Republican</span> <span m="2509770">or</span> <span m="2511060">you</span> <span m="2511220">were</span> <span m="2511310">unlucky</span> <span m="2511920">in</span> <span m="2512010">the</span> <span m="2512090">1%</span> <span m="2512425">you</span> <span m="2512760">sample.</span> <span m="2514740">That's</span> <span m="2515010">what</span> <span m="2515120">you</span> <span m="2515220">can</span> <span m="2515340">conclude</span> <span m="2517760">because</span> <span m="2518080">the</span> <span m="2518170">population</span> <span m="2518870">really</span> <span m="2519120">is</span> <span m="2519290">fixed</span> <span m="2519640">in</span> <span m="2519700">this</span> <span m="2519870">case.</span> </p>
<p><span m="2520110">It</span> <span m="2520190">is</span> <span m="2520380">what</span> <span m="2520530">it</span> <span m="2520620">is.</span> <span m="2521060">There's</span> <span m="2521240">no</span> <span m="2521460">randomness</span> <span m="2522000">in the</span> <span m="2522100">population.</span> <span m="2524340">All</span> <span m="2524370">right?</span> <span m="2524590">So</span> <span m="2524670">you</span> <span m="2524830">have</span> <span m="2525160">next</span> <span m="2525430">week</span> <span m="2525880">for</span> <span m="2526000">recitation.</span> <span m="2527630">You're</span> <span m="2527800">going</span> <span m="2527910">to</span> <span m="2527960">design</span> <span m="2528340">a</span> <span m="2528410">pole</span> <span m="2529180">and</span> <span m="2529360">work</span> <span m="2529610">through</span> <span m="2529770">how</span> <span m="2529890">to</span> <span m="2529980">calculate</span> <span m="2530390">the</span> <span m="2530470">margin</span> <span m="2530830">of</span> <span m="2530910">error</span> <span m="2531300">and</span> <span m="2531400">work</span> <span m="2531640">through</span> <span m="2531760">what</span> <span m="2531990">that</span> <span m="2532180">really</span> <span m="2532420">means</span> <span m="2533950">in</span> <span m="2534090">terms</span> <span m="2534640">of</span> <span m="2535270">what</span> <span m="2535410">the</span> <span m="2535500">population</span> <span m="2536080">is</span> <span m="2536210">like.</span> </p>
<p><span m="2537410">Now</span> <span m="2537550">of</span> <span m="2537630">course,</span> <span m="2538380">if</span> <span m="2538480">it</span> <span m="2538560">comes</span> <span m="2538860">out</span> <span m="2539220">100</span> <span m="2539550">straight</span> <span m="2539850">times</span> <span m="2540160">heads,</span> <span m="2540530">you've</span> <span m="2540620">got</span> <span m="2540710">to</span> <span m="2540760">be</span> <span m="2540870">really</span> <span m="2541300">unlucky</span> <span m="2542330">to</span> <span m="2542450">have</span> <span m="2542610">the</span> <span m="2542690">fair</span> <span m="2542920">coin.</span> <span m="2543590">And</span> <span m="2543640">the</span> <span m="2543720">same</span> <span m="2543980">thing</span> <span m="2544140">with</span> <span m="2544240">designing</span> <span m="2544580">the</span> <span m="2544660">poll</span> <span m="2545460">if</span> <span m="2545610">you're</span> <span m="2545720">way</span> <span m="2545910">off.</span> <span m="2548270">Any</span> <span m="2548420">questions</span> <span m="2550160">about</span> <span m="2550430">that?</span> </p>
<p><span m="2552990">OK.</span> <span m="2553260">The</span> <span m="2553370">next</span> <span m="2553680">example</span> <span m="2554700">comes</span> <span m="2554940">up</span> <span m="2555140">all</span> <span m="2555360">the</span> <span m="2555470">time</span> <span m="2556380">in</span> <span m="2556530">practice.</span> <span m="2557970">And</span> <span m="2558150">that's</span> <span m="2558410">with</span> <span m="2558610">medical</span> <span m="2559130">testing.</span> <span m="2560946">Maybe</span> <span m="2561410">I'll</span> <span m="2561560">leave--</span> <span m="2563346">no.</span> <span m="2563820">I'll take</span> <span m="2564100">that down.</span> <span m="2564380">We</span> <span m="2564640">know</span> <span m="2565137">that now.</span> </p>
<p></p>
<p><span m="2585030">Now</span> <span m="2585280">in</span> <span m="2585350">this</span> <span m="2585560">case--</span> <span m="2588040">in</span> <span m="2588110">fact,</span> <span m="2588290">this</span> <span m="2588470">is</span> <span m="2588520">a</span> <span m="2588580">question</span> <span m="2588990">we</span> <span m="2589090">had</span> <span m="2589260">on</span> <span m="2589360">the</span> <span m="2589440">final</span> <span m="2589780">exam</span> <span m="2590200">a</span> <span m="2590250">few</span> <span m="2590440">years</span> <span m="2590710">ago.</span> <span m="2591530">And</span> <span m="2591740">there's</span> <span m="2591900">a</span> <span m="2591970">good</span> <span m="2592240">chance</span> <span m="2593550">this kind</span> <span m="2593730">of</span> <span m="2593780">question's</span> <span m="2594230">going</span> <span m="2594330">to</span> <span m="2594380">be</span> <span m="2594500">on</span> <span m="2594580">the</span> <span m="2594650">final</span> <span m="2594960">this</span> <span m="2595170">year.</span> <span m="2596340">There's</span> <span m="2596960">a</span> <span m="2597030">disease</span> <span m="2597480">out</span> <span m="2597740">there.</span> <span m="2598760">And</span> <span m="2598970">you</span> <span m="2599050">can</span> <span m="2599160">have</span> <span m="2599330">a</span> <span m="2599400">test</span> <span m="2599750">for</span> <span m="2599990">it.</span> </p>
<p><span m="2601390">But</span> <span m="2601570">like</span> <span m="2601760">most</span> <span m="2602180">medical</span> <span m="2602570">tests,</span> <span m="2602930">they're</span> <span m="2603080">not</span> <span m="2603510">perfect.</span> <span m="2604570">Sometimes</span> <span m="2604860">when</span> <span m="2605250">it</span> <span m="2605330">says</span> <span m="2605540">you've</span> <span m="2605680">got</span> <span m="2605900">the</span> <span m="2605970">disease</span> <span m="2606450">you</span> <span m="2606590">really</span> <span m="2606840">don't.</span> <span m="2607900">And</span> <span m="2608040">if it</span> <span m="2608080">ways</span> <span m="2608340">you</span> <span m="2608450">don't</span> <span m="2608740">have</span> <span m="2608990">it,</span> <span m="2609410">you</span> <span m="2609570">really</span> <span m="2609870">do.</span> </p>
<p><span m="2611840">So</span> <span m="2612030">in</span> <span m="2612090">this</span> <span m="2612280">case,</span> <span m="2612650">we're</span> <span m="2612670">going</span> <span m="2612780">to</span> <span m="2612820">assume</span> <span m="2613240">that</span> <span m="2613420">10%</span> <span m="2613980">of</span> <span m="2614020">the</span> <span m="2614110">population</span> <span m="2619100">has</span> <span m="2619510">the</span> <span m="2619610">disease,</span> <span m="2622960">whatever</span> <span m="2623590">it</span> <span m="2623690">is.</span> <span m="2626000">You</span> <span m="2626140">don't</span> <span m="2626310">get</span> <span m="2626460">symptoms</span> <span m="2626880">right</span> <span m="2627090">away.</span> <span m="2627330">So</span> <span m="2627450">you</span> <span m="2627520">have</span> <span m="2627700">this</span> <span m="2627920">test.</span> </p>
<p><span m="2629190">But</span> <span m="2629710">if</span> <span m="2629960">you</span> <span m="2630130">have</span> <span m="2630550">the</span> <span m="2630640">disease</span> <span m="2637840">there</span> <span m="2638050">is</span> <span m="2638140">a</span> <span m="2638230">10%</span> <span m="2638990">chance</span> <span m="2644410">that the</span> <span m="2644430">test</span> <span m="2644850">is</span> <span m="2644970">negative.</span> <span m="2648940">And</span> <span m="2649130">this</span> <span m="2649280">is</span> <span m="2649400">called</span> <span m="2649790">a</span> <span m="2649870">false</span> <span m="2650310">negative,</span> <span m="2655040">because</span> <span m="2655340">the</span> <span m="2655450">test</span> <span m="2655710">comes</span> <span m="2655930">back</span> <span m="2656180">negative</span> <span m="2656630">but</span> <span m="2656750">it's</span> <span m="2657030">wrong,</span> <span m="2657410">because</span> <span m="2657580">you</span> <span m="2657680">have</span> <span m="2657890">the</span> <span m="2657960">disease.</span> </p>
<p><span m="2659750">And</span> <span m="2659960">similarly,</span> <span m="2660960">if</span> <span m="2661140">you</span> <span m="2661250">have</span> <span m="2661560">the</span> <span m="2661640">disease--</span> <span m="2662620">or</span> <span m="2662780">sorry--</span> <span m="2662880">if</span> <span m="2662970">you</span> <span m="2663100">don't</span> <span m="2663450">have</span> <span m="2663680">the</span> <span m="2663750">disease,</span> <span m="2669240">there's</span> <span m="2669480">a</span> <span m="2669580">30%</span> <span m="2670350">chance</span> <span m="2672380">that</span> <span m="2672760">the</span> <span m="2672850">test</span> <span m="2673170">comes</span> <span m="2673390">back</span> <span m="2673670">positive.</span> <span m="2675170">And</span> <span m="2675300">it's</span> <span m="2675420">called</span> <span m="2675700">a</span> <span m="2675760">false</span> <span m="2676150">positive</span> <span m="2679000">because</span> <span m="2679800">it</span> <span m="2679890">came</span> <span m="2680190">back</span> <span m="2680430">positive,</span> <span m="2681560">but</span> <span m="2681720">you</span> <span m="2681820">don't</span> <span m="2682080">have</span> <span m="2682370">it.</span> <span m="2683870">So</span> <span m="2683990">the</span> <span m="2684090">test</span> <span m="2684400">is</span> <span m="2684520">pretty</span> <span m="2684800">good.</span> <span m="2685800">Right?</span> <span m="2686500">It's</span> <span m="2686900">10%</span> <span m="2688040">false</span> <span m="2688400">negative</span> <span m="2688720">right,</span> <span m="2689080">30%</span> <span m="2689720">false</span> <span m="2690020">positive</span> <span m="2690480">right.</span> </p>
<p><span m="2692390">Now</span> <span m="2692980">say</span> <span m="2693150">you</span> <span m="2693310">select</span> <span m="2693720">a</span> <span m="2693790">random</span> <span m="2694100">person</span> <span m="2695970">and</span> <span m="2696220">they</span> <span m="2696330">test</span> <span m="2696750">positive.</span> <span m="2699210">What</span> <span m="2699390">you</span> <span m="2699520">want</span> <span m="2699730">to</span> <span m="2699780">know</span> <span m="2700250">is</span> <span m="2700390">the</span> <span m="2700480">probability</span> <span m="2701120">they</span> <span m="2701250">have</span> <span m="2701490">the</span> <span m="2701590">disease</span> <span m="2702950">given</span> <span m="2703140">that</span> <span m="2703230">it's</span> <span m="2703370">a</span> <span m="2703450">random</span> <span m="2703550">person.</span> <span m="2707730">So</span> <span m="2709910">actually,</span> <span m="2710180">this</span> <span m="2710470">came</span> <span m="2710730">up</span> <span m="2710940">in</span> <span m="2711060">my</span> <span m="2711260">personal</span> <span m="2711780">life.</span> </p>
<p><span m="2712630">Many</span> <span m="2713120">years</span> <span m="2713400">ago</span> <span m="2714330">when</span> <span m="2714600">my</span> <span m="2714710">wife</span> <span m="2715020">was</span> <span m="2715170">pregnant</span> <span m="2716190">with</span> <span m="2717000">Alex,</span> <span m="2718080">she</span> <span m="2718290">was</span> <span m="2718480">exposed</span> <span m="2719220">to</span> <span m="2719280">somebody</span> <span m="2719770">with</span> <span m="2720350">TB</span> <span m="2721160">here</span> <span m="2721460">at</span> <span m="2721510">MIT.</span> <span m="2722840">And</span> <span m="2723410">she</span> <span m="2723560">took</span> <span m="2723740">the</span> <span m="2723850">test.</span> <span m="2725400">And</span> <span m="2725630">it</span> <span m="2725700">came</span> <span m="2725920">back</span> <span m="2726120">positive.</span> </p>
<p><span m="2728040">Now</span> <span m="2728920">the</span> <span m="2729020">bad</span> <span m="2729280">thing--</span> <span m="2729710">TB's</span> <span m="2730130">a</span> <span m="2730180">bad</span> <span m="2730500">thing.</span> <span m="2730760">You</span> <span m="2730860">don't</span> <span m="2730960">want</span> <span m="2730980">to</span> <span m="2731030">get</span> <span m="2731250">it.</span> <span m="2731680">But</span> <span m="2731920">the</span> <span m="2732520">medicine</span> <span m="2733180">for</span> <span m="2733525">it</span> <span m="2734500">you</span> <span m="2734650">take</span> <span m="2734940">for</span> <span m="2735050">six</span> <span m="2735350">months.</span> <span m="2735990">And</span> <span m="2736150">she</span> <span m="2736270">was</span> <span m="2736440">worried</span> <span m="2736860">about</span> <span m="2737120">taking</span> <span m="2737400">medicine</span> <span m="2737760">for</span> <span m="2737870">six</span> <span m="2738130">months</span> <span m="2738400">when</span> <span m="2738510">she's</span> <span m="2738710">pregnant</span> <span m="2740130">because</span> <span m="2740340">who</span> <span m="2740530">knows</span> <span m="2741100">what</span> <span m="2741210">the</span> <span m="2741300">TB</span> <span m="2741730">medicine</span> <span m="2742270">does</span> <span m="2743040">kind</span> <span m="2743210">of</span> <span m="2743310">thing</span> <span m="2743830">if</span> <span m="2743910">you</span> <span m="2744210">have</span> <span m="2744390">a</span> <span m="2744440">baby.</span> </p>
<p><span m="2745940">So</span> <span m="2746120">she</span> <span m="2746330">asked</span> <span m="2746530">the</span> <span m="2746620">doc,</span> <span m="2746970">what's</span> <span m="2747290">the</span> <span m="2747390">probability</span> <span m="2747970">I</span> <span m="2748070">really</span> <span m="2748440">have</span> <span m="2748710">the</span> <span m="2748780">disease?</span> <span m="2751320">The</span> <span m="2751380">doc</span> <span m="2751655">doesn't</span> <span m="2751930">know.</span> <span m="2753570">The doc</span> <span m="2753780">maybe</span> <span m="2754100">could</span> <span m="2754310">give</span> <span m="2754440">you</span> <span m="2754570">some</span> <span m="2754740">of</span> <span m="2754810">these</span> <span m="2755020">steps,</span> <span m="2755620">10%</span> <span m="2756460">false</span> <span m="2756730">negative,</span> <span m="2757060">30%</span> <span m="2757590">false</span> <span m="2757850">positive.</span> <span m="2758350">But</span> <span m="2758660">it</span> <span m="2758820">tested</span> <span m="2759170">positive.</span> <span m="2759760">So</span> <span m="2759790">they</span> <span m="2759940">just</span> <span m="2760120">normally</span> <span m="2760920">give</span> <span m="2761100">you</span> <span m="2761890">the</span> <span m="2761980">medicine.</span> </p>
<p><span m="2763530">So</span> <span m="2763920">say</span> <span m="2764090">this</span> <span m="2764300">was</span> <span m="2764480">the</span> <span m="2764620">story.</span> <span m="2765140">What</span> <span m="2765410">would</span> <span m="2765560">you</span> <span m="2765750">say?</span> <span m="2766490">What</span> <span m="2766590">do</span> <span m="2766630">you</span> <span m="2766740">think?</span> <span m="2767670">How</span> <span m="2767770">many</span> <span m="2767950">people</span> <span m="2768260">think</span> <span m="2768510">that it's</span> <span m="2768730">a least</span> <span m="2768800">a</span> <span m="2769110">70%</span> <span m="2769800">chance</span> <span m="2770090">you</span> <span m="2770160">got</span> <span m="2770320">the</span> <span m="2770400">disease?</span> <span m="2772900">She</span> <span m="2773000">tested</span> <span m="2773380">positive</span> <span m="2773940">and</span> <span m="2774030">it's</span> <span m="2774700">only</span> <span m="2774930">got</span> <span m="2775080">a</span> <span m="2775130">30%</span> <span m="2775990">false</span> <span m="2776240">positive</span> <span m="2776600">rate.</span> <span m="2776820">Anybody?</span> <span m="2778701">So</span> <span m="2779140">you</span> <span m="2779230">don't</span> <span m="2779380">think</span> <span m="2779530">she's</span> <span m="2779780">likely</span> <span m="2780080">to</span> <span m="2780170">have</span> <span m="2780440">it.</span> </p>
<p><span m="2781260">How</span> <span m="2781370">many</span> <span m="2781520">people</span> <span m="2781780">think</span> <span m="2781960">it's</span> <span m="2782110">better</span> <span m="2782330">than</span> <span m="2782490">50-50</span> <span m="2783720">you</span> <span m="2783820">have</span> <span m="2784030">the</span> <span m="2784110">disease?</span> <span m="2786820">A</span> <span m="2787060">few.</span> <span m="2787870">How</span> <span m="2788010">many</span> <span m="2788210">people</span> <span m="2788570">think</span> <span m="2788930">less</span> <span m="2789210">than</span> <span m="2789340">50%.</span> <span m="2791420">A</span> <span m="2791490">bunch.</span> <span m="2792290">Yeah.</span> <span m="2793070">You're</span> <span m="2793260">right,</span> <span m="2793740">in</span> <span m="2793850">fact.</span> </p>
<p><span m="2796250">Let's</span> <span m="2796440">figure</span> <span m="2796690">out</span> <span m="2796790">the</span> <span m="2796950">answer.</span> <span m="2797440">It's</span> <span m="2797530">easy</span> <span m="2797780">to</span> <span m="2797850">do.</span> <span m="2799550">So</span> <span m="2799810">A</span> <span m="2800150">is</span> <span m="2800370">the</span> <span m="2800560">event</span> <span m="2802770">the</span> <span m="2802840">person</span> <span m="2803290">has</span> <span m="2803580">the</span> <span m="2803660">disease.</span> <span m="2811800">And</span> <span m="2812260">B</span> <span m="2813490">is</span> <span m="2813750">the</span> <span m="2814010">event</span> <span m="2815100">that</span> <span m="2815440">the</span> <span m="2815620">person</span> <span m="2816220">tests</span> <span m="2816560">positive.</span> </p>
<p><span m="2821120">And</span> <span m="2821370">of</span> <span m="2821440">course</span> <span m="2821690">what we</span> <span m="2821850">want</span> <span m="2822070">to</span> <span m="2822120">know</span> <span m="2823400">is</span> <span m="2823670">the</span> <span m="2823760">probability</span> <span m="2824410">you</span> <span m="2824500">have</span> <span m="2824790">the</span> <span m="2824870">disease</span> <span m="2825390">given</span> <span m="2825670">that</span> <span m="2825730">you</span> <span m="2825830">tested</span> <span m="2826250">positive.</span> <span m="2827550">And</span> <span m="2827690">that's</span> <span m="2827960">just</span> <span m="2828410">the</span> <span m="2829150">probability</span> <span m="2829660">of</span> <span m="2829720">both</span> <span m="2830020">events</span> <span m="2830990">divided</span> <span m="2831420">by</span> <span m="2831550">the</span> <span m="2831660">probability</span> <span m="2832890">of</span> <span m="2834080">testing</span> <span m="2834440">positive.</span> <span m="2836340">So</span> <span m="2836390">let's</span> <span m="2836610">figure</span> <span m="2836860">that</span> <span m="2837100">out</span> <span m="2839030">by</span> <span m="2839140">drawing</span> <span m="2839500">the</span> <span m="2839590">tree.</span> </p>
<p></p>
<p><span m="2858750">So</span> <span m="2859370">first,</span> <span m="2860090">do you</span> <span m="2860210">have</span> <span m="2860490">the</span> <span m="2860570">disease?</span> <span m="2863980">And</span> <span m="2864440">it's</span> <span m="2864600">yes</span> <span m="2864950">or</span> <span m="2865040">no.</span> <span m="2869320">And</span> <span m="2870140">let's</span> <span m="2870310">see.</span> <span m="2870380">The</span> <span m="2870480">probability</span> <span m="2871160">of</span> <span m="2871210">having</span> <span m="2871630">the</span> <span m="2871720">disease,</span> <span m="2873080">what</span> <span m="2873250">is</span> <span m="2873360">that</span> <span m="2873690">for</span> <span m="2873850">a</span> <span m="2873920">random</span> <span m="2874290">person?</span> <span m="2876010">10%.</span> <span m="2876810">that</span> <span m="2877000">the stat.</span> <span m="2878380">So</span> <span m="2878530">it's--</span> <span m="2878820">actually,</span> <span m="2879190">we'll call it</span> <span m="2879490">0.1.</span> <span m="2881274">And</span> <span m="2881720">9.9</span> <span m="2882390">you</span> <span m="2882490">don't</span> <span m="2882750">have</span> <span m="2883020">it.</span> </p>
<p><span m="2884130">And</span> <span m="2884340">then</span> <span m="2884490">there's</span> <span m="2884680">the</span> <span m="2884800">test.</span> <span m="2887940">Well,</span> <span m="2888180">you</span> <span m="2888330">can</span> <span m="2888460">be</span> <span m="2888950">positive</span> <span m="2889600">or</span> <span m="2889670">negative.</span> <span m="2894360">Now</span> <span m="2895060">if</span> <span m="2895310">you</span> <span m="2895560">have</span> <span m="2895850">the</span> <span m="2895940">disease,</span> <span m="2899000">there</span> <span m="2899520">is</span> <span m="2899820">a--</span> <span m="2900485">the</span> <span m="2900970">chance</span> <span m="2901480">you</span> <span m="2901620">test</span> <span m="2902060">negative</span> <span m="2903020">is</span> <span m="2903290">10%,</span> <span m="2905520">0.1.</span> <span m="2906880">Therefore</span> <span m="2907430">there's a</span> <span m="2907670">90%</span> <span m="2908430">chance</span> <span m="2908740">you</span> <span m="2908840">test</span> <span m="2909120">positive.</span> </p>
<p><span m="2911450">Now</span> <span m="2911620">if,</span> <span m="2911690">you</span> <span m="2911780">don't</span> <span m="2912090">have</span> <span m="2912320">the</span> <span m="2912390">disease,</span> <span m="2912810">you</span> <span m="2912920">could</span> <span m="2913070">test</span> <span m="2913370">either</span> <span m="2913560">way.</span> <span m="2916740">If</span> <span m="2916860">you</span> <span m="2916980">don't</span> <span m="2917420">have</span> <span m="2917660">the</span> <span m="2917740">disease</span> <span m="2918930">there's</span> <span m="2919120">a</span> <span m="2919200">30%</span> <span m="2920070">chance</span> <span m="2920390">you</span> <span m="2920480">test</span> <span m="2920790">positive.</span> <span m="2922590">30</span> <span m="2923230">here</span> <span m="2923520">and</span> <span m="2923670">70%</span> <span m="2924100">percent</span> <span m="2924460">chance</span> <span m="2924800">you're</span> <span m="2924870">negative.</span> </p>
<p><span m="2927760">Now</span> <span m="2928050">we</span> <span m="2928170">can</span> <span m="2928320">compute</span> <span m="2929010">each</span> <span m="2929280">sample</span> <span m="2929670">point</span> <span m="2929910">probability.</span> <span m="2931620">This</span> <span m="2931850">one</span> <span m="2932090">is</span> <span m="2932600">0.1</span> <span m="2933170">times</span> <span m="2933410">0.9</span> <span m="2933960">is</span> <span m="2934120">0.09.</span> <span m="2936560">0.1</span> <span m="2937030">times</span> <span m="2937340">1</span> <span m="2937620">is</span> <span m="2937750">0.01.</span> <span m="2940050">0.9</span> <span m="2940325">and</span> <span m="2940600">0.3</span> <span m="2941200">is</span> <span m="2941520">0.27.</span> <span m="2943720">0.9</span> <span m="2944075">and</span> <span m="2944430">0.7</span> <span m="2945220">is</span> <span m="2945630">0.63.</span> </p>
<p><span m="2949580">So</span> <span m="2949720">all</span> <span m="2950030">sample</span> <span m="2950390">points</span> <span m="2950690">are</span> <span m="2950750">figured</span> <span m="2951050">out.</span> <span m="2952350">Now</span> <span m="2952730">we</span> <span m="2953000">figure</span> <span m="2953310">out</span> <span m="2953780">which</span> <span m="2954000">sample</span> <span m="2954340">points</span> <span m="2954580">are</span> <span m="2954780">in which</span> <span m="2954990">sets.</span> <span m="2956240">So we</span> <span m="2956390">have event</span> <span m="2956700">A,</span> <span m="2957020">event</span> <span m="2957360">B,</span> <span m="2958750">and</span> <span m="2958900">A</span> <span m="2959100">intersect</span> <span m="2959580">B.</span> <span m="2961170">Let's</span> <span m="2961380">see.</span> <span m="2961650">A</span> <span m="2961920">is</span> <span m="2962080">the</span> <span m="2962230">event</span> <span m="2962750">you</span> <span m="2962840">have</span> <span m="2963220">the</span> <span m="2963300">disease.</span> <span m="2965300">That's</span> <span m="2965540">these</span> <span m="2965770">guys.</span> <span m="2967590">B</span> <span m="2967850">is</span> <span m="2967990">the</span> <span m="2968140">event</span> <span m="2969260">you</span> <span m="2969430">test</span> <span m="2969870">positive.</span> <span m="2971790">That's</span> <span m="2972070">this</span> <span m="2972270">one</span> <span m="2973860">and</span> <span m="2974970">this</span> <span m="2975160">one.</span> <span m="2976840">A</span> <span m="2977060">intersect</span> <span m="2977580">B</span> <span m="2978330">is</span> <span m="2978470">just</span> <span m="2978710">this</span> <span m="2978890">one.</span> </p>
<p><span m="2980616">All right.</span> <span m="2981020">We're</span> <span m="2981180">almost</span> <span m="2981440">done.</span> <span m="2983310">Let's</span> <span m="2983430">just</span> <span m="2983630">figure</span> <span m="2983910">out</span> <span m="2984050">the</span> <span m="2984130">probability</span> <span m="2984690">you</span> <span m="2984780">have</span> <span m="2985040">the</span> <span m="2985110">disease.</span> <span m="2987190">What's</span> <span m="2987290">the</span> <span m="2987380">probability</span> <span m="2987820">of</span> <span m="2987890">A</span> <span m="2988050">intersect</span> <span m="2988520">B?</span> <span m="2992340">0.09.</span> <span m="2993120">It's</span> <span m="2993230">just</span> <span m="2993390">that</span> <span m="2993560">one</span> <span m="2993780">sample</span> <span m="2994170">point.</span> </p>
<p><span m="2996740">What's</span> <span m="2997090">the</span> <span m="2997200">probability</span> <span m="2998470">that</span> <span m="2998650">you</span> <span m="2998760">tested</span> <span m="2999330">positive?</span> <span m="3002170">0.36.</span> <span m="3003630">Yeah.</span> <span m="3003730">0.09</span> <span m="3004590">plus</span> <span m="3004850">0.27,</span> <span m="3011650">which</span> <span m="3011860">is</span> <span m="3011970">0.36.</span> <span m="3013610">So</span> <span m="3013640">I</span> <span m="3013720">got</span> <span m="3013860">0.09</span> <span m="3014852">over</span> <span m="3015660">0.36</span> <span m="3017850">is</span> <span m="3018020">1/4.</span> <span m="3021430">Wow.</span> <span m="3022790">That</span> <span m="3023190">seems</span> <span m="3023690">bizarre.</span> </p>
<p><span m="3024300">Right?</span> <span m="3024570">You've</span> <span m="3024680">got</span> <span m="3024830">a</span> <span m="3024910">test,</span> <span m="3027260">10%</span> <span m="3027530">percent</span> <span m="3027920">false</span> <span m="3028240">negative,</span> <span m="3028740">30%</span> <span m="3029390">false</span> <span m="3029670">positive.</span> <span m="3030250">Yet,</span> <span m="3031010">when</span> <span m="3031140">you</span> <span m="3031230">test</span> <span m="3031560">positive</span> <span m="3032030">there's</span> <span m="3032330">only</span> <span m="3032890">a</span> <span m="3032950">25%</span> <span m="3033790">chance</span> <span m="3034060">you</span> <span m="3034140">have</span> <span m="3034320">the</span> <span m="3034390">disease.</span> <span m="3036490">So</span> <span m="3036640">maybe</span> <span m="3036870">you</span> <span m="3036930">don't</span> <span m="3037160">take</span> <span m="3037320">the</span> <span m="3037390">medicine.</span> <span m="3038610">So</span> <span m="3038790">if there's</span> <span m="3039050">risk</span> <span m="3039320">both</span> <span m="3039540">ways,</span> <span m="3040360">probably</span> <span m="3040830">don't</span> <span m="3041150">have</span> <span m="3041360">the</span> <span m="3041420">disease.</span> <span m="3042930">Yeah?</span> </p>
<p><span m="3044970">AUDIENCE: [INAUDIBLE]</span> <span m="3045470">disease</span> <span m="3045970">change</span> <span m="3046470">because</span> <span m="3046970">you've</span> <span m="3047470">already been</span> <span m="3047970">exposed to somebody</span> <span m="3048470">that has it?</span> </p>
<p><span m="3048970">PROFESSOR: That's</span> <span m="3049255">a</span> <span m="3049540">great</span> <span m="3050000">point,</span> <span m="3050860">great</span> <span m="3051280">point,</span> <span m="3052530">because</span> <span m="3053110">there's</span> <span m="3053300">additional</span> <span m="3053810">information</span> <span m="3054440">conditioning</span> <span m="3054990">this</span> <span m="3055450">in</span> <span m="3055570">the</span> <span m="3055690">personal</span> <span m="3056060">example</span> <span m="3056440">I</span> <span m="3056510">cited.</span> <span m="3057190">You</span> <span m="3057370">were</span> <span m="3057480">exposed</span> <span m="3058050">to</span> <span m="3058150">somebody.</span> <span m="3059250">So</span> <span m="3059520">we</span> <span m="3059620">need</span> <span m="3059780">to</span> <span m="3059850">condition</span> <span m="3060280">on</span> <span m="3060390">that</span> <span m="3060570">as</span> <span m="3060680">well,</span> <span m="3061900">which</span> <span m="3062030">raises</span> <span m="3062650">the</span> <span m="3062770">chance</span> <span m="3063190">you</span> <span m="3063300">have</span> <span m="3063530">the</span> <span m="3063570">disease.</span> <span m="3064250">That's</span> <span m="3064460">a</span> <span m="3064510">great</span> <span m="3064760">point.</span> <span m="3066040">Yeah.</span> </p>
<p><span m="3067950">Just</span> <span m="3068170">like</span> <span m="3068340">in</span> <span m="3068440">the--</span> <span m="3069030">well,</span> <span m="3069190">we</span> <span m="3069280">haven't</span> <span m="3069790">got</span> <span m="3069910">to that</span> <span m="3069980">example.</span> <span m="3070420">Do</span> <span m="3070500">another</span> <span m="3070730">example</span> <span m="3071090">with</span> <span m="3071210">that</span> <span m="3071410">exact</span> <span m="3071990">kind</span> <span m="3072170">of</span> <span m="3072270">thing</span> <span m="3072420">is</span> <span m="3072550">very</span> <span m="3072730">important.</span> <span m="3074486">All right.</span> <span m="3074930">So</span> <span m="3075240">this</span> <span m="3075370">is</span> <span m="3075470">sort</span> <span m="3075710">of</span> <span m="3075920">paradoxical</span> <span m="3077680">that</span> <span m="3077810">it</span> <span m="3077920">looks</span> <span m="3078130">like</span> <span m="3078250">a</span> <span m="3078310">pretty</span> <span m="3078530">good</span> <span m="3078720">test--</span> <span m="3080040">low</span> <span m="3080330">false</span> <span m="3080650">positive,</span> <span m="3081120">full</span> <span m="3081380">false</span> <span m="3081700">negatives,</span> <span m="3082100">but</span> <span m="3083570">likely</span> <span m="3083930">be</span> <span m="3084050">wrong,</span> <span m="3084870">at</span> <span m="3084990">least</span> <span m="3085250">if</span> <span m="3085350">it</span> <span m="3085460">tells</span> <span m="3085690">you</span> <span m="3086000">have</span> <span m="3086320">the</span> <span m="3086400">disease.</span> </p>
<p><span m="3088716">In fact,</span> <span m="3089210">let's</span> <span m="3089380">figure</span> <span m="3089620">out.</span> <span m="3089720">What's</span> <span m="3089900">the</span> <span m="3089990">probability</span> <span m="3090480">that the</span> <span m="3090580">test</span> <span m="3090900">is</span> <span m="3091020">correct?</span> <span m="3095620">What's</span> <span m="3095810">the</span> <span m="3095900">probability</span> <span m="3097100">the</span> <span m="3097190">test</span> <span m="3097550">is</span> <span m="3097660">right</span> <span m="3098520">in</span> <span m="3098640">general?</span> <span m="3103030">72%.</span> <span m="3104350">Let's</span> <span m="3104580">see.</span> <span m="3105570">So</span> <span m="3105790">it would</span> <span m="3105910">be</span> <span m="3106070">0.09</span> <span m="3107790">plus</span> <span m="3107980">0.63.</span> <span m="3108814">72%.</span> </p>
<p><span m="3116780">So</span> <span m="3117020">it's</span> <span m="3117130">likely</span> <span m="3117490">to</span> <span m="3117560">be</span> <span m="3117680">right.</span> <span m="3119890">But</span> <span m="3120050">if</span> <span m="3120130">it</span> <span m="3120170">tells</span> <span m="3120420">you</span> <span m="3120660">you have</span> <span m="3120830">the</span> <span m="3120900">disease</span> <span m="3121370">it's</span> <span m="3121530">likely</span> <span m="3121830">to</span> <span m="3121880">be</span> <span m="3122000">wrong.</span> <span m="3125500">It's</span> <span m="3125610">hard.</span> <span m="3126850">Why</span> <span m="3127060">is</span> <span m="3127230">this</span> <span m="3127550">happening?</span> <span m="3128930">Why</span> <span m="3129050">does</span> <span m="3129190">it</span> <span m="3129480">come out that</span> <span m="3129640">way?</span> <span m="3129780">Yeah?</span> </p>
<p><span m="3132980">AUDIENCE: Then</span> <span m="3133270">there</span> <span m="3133670">is</span> <span m="3134280">only</span> <span m="3134600">a</span> <span m="3134880">1</span> <span m="3135160">in</span> <span m="3135230">64</span> <span m="3135780">chance</span> <span m="3136190">that</span> <span m="3136630">you</span> <span m="3137060">have the</span> <span m="3137360">disease.</span> <span m="3138630">So</span> <span m="3139040">if it comes</span> <span m="3139350">back</span> <span m="3139540">negative,</span> <span m="3140420">then</span> <span m="3140710">it's a</span> <span m="3140870">pretty</span> <span m="3141060">good</span> <span m="3141180">indication</span> <span m="3141720">that</span> <span m="3141890">you're</span> <span m="3142030">OK.</span> </p>
<p><span m="3142770">PROFESSOR: Yeah.</span> <span m="3143250">If it comes back</span> <span m="3143730">negative</span> <span m="3144080">than</span> <span m="3144390">it really</span> <span m="3144760">is</span> <span m="3145920">doing</span> <span m="3146020">very</span> <span m="3146260">well.</span> <span m="3146980">That's</span> <span m="3147240">right.</span> <span m="3148080">But</span> <span m="3148270">why</span> <span m="3148520">is</span> <span m="3148640">it</span> <span m="3148710">when</span> <span m="3148820">it</span> <span m="3148870">comes</span> <span m="3149080">back</span> <span m="3149300">positive</span> <span m="3149615">that</span> <span m="3149930">you're</span> <span m="3150210">unlikely</span> <span m="3150590">to</span> <span m="3150690">have</span> <span m="3150880">the</span> <span m="3150940">disease</span> <span m="3151320">if</span> <span m="3151410">it's</span> <span m="3151510">a</span> <span m="3151570">good</span> <span m="3151750">test.</span> <span m="3152070">Yeah.</span> </p>
<p><span m="3153340">AUDIENCE: The</span> <span m="3153800">disease is so rare.</span> </p>
<p><span m="3154260">PROFESSOR: The disease</span> <span m="3154610">is</span> <span m="3154960">so</span> <span m="3155110">rare.</span> <span m="3155760">Absolutely.</span> <span m="3157500">This</span> <span m="3157760">number</span> <span m="3158150">here</span> <span m="3158500">is</span> <span m="3158570">so</span> <span m="3158910">small.</span> <span m="3159770">And</span> <span m="3160340">that's</span> <span m="3160560">what's</span> <span m="3160740">doing</span> <span m="3161060">it.</span> <span m="3161960">Because</span> <span m="3162270">if</span> <span m="3162340">you</span> <span m="3162480">look</span> <span m="3162710">at</span> <span m="3162830">how</span> <span m="3163030">many</span> <span m="3163280">people</span> <span m="3164470">have</span> <span m="3164680">the</span> <span m="3164750">disease</span> <span m="3165020">and</span> <span m="3165110">test</span> <span m="3165310">positive,</span> <span m="3165770">it's</span> <span m="3165910">0.09.</span> <span m="3167120">So</span> <span m="3167550">many</span> <span m="3167820">people</span> <span m="3168210">don't</span> <span m="3168600">have</span> <span m="3168850">the</span> <span m="3168910">disease</span> <span m="3170270">that even</span> <span m="3170600">with</span> <span m="3170740">a</span> <span m="3170790">small</span> <span m="3171140">false</span> <span m="3171450">positive</span> <span m="3171910">rate,</span> <span m="3172590">this</span> <span m="3172810">number</span> <span m="3173120">swamps</span> <span m="3173570">out</span> <span m="3174080">that</span> <span m="3174310">number.</span> </p>
<p><span m="3175380">In</span> <span m="3175490">fact,</span> <span m="3175700">imagine</span> <span m="3176120">nobody</span> <span m="3176540">had</span> <span m="3176750">the</span> <span m="3176820">disease.</span> <span m="3177930">You'd</span> <span m="3178030">have</span> <span m="3178220">a</span> <span m="3178260">0</span> <span m="3178690">here.</span> <span m="3180420">All</span> <span m="3180850">right?</span> <span m="3181140">And</span> <span m="3181230">then</span> <span m="3182080">you</span> <span m="3182120">would</span> <span m="3182210">always</span> <span m="3182470">be</span> <span m="3182580">wrong</span> <span m="3182990">if you</span> <span m="3183080">said</span> <span m="3183300">you had it.</span> <span m="3184750">OK?</span> <span m="3185480">That's</span> <span m="3186290">good.</span> </p>
<p><span m="3189020">OK.</span> <span m="3189280">This</span> <span m="3189500">comes</span> <span m="3189700">up</span> <span m="3189800">in</span> <span m="3189900">weather</span> <span m="3190230">prediction,</span> <span m="3190650">the</span> <span m="3190740">same</span> <span m="3191060">paradox.</span> <span m="3191980">For</span> <span m="3192090">example,</span> <span m="3193210">say</span> <span m="3193340">you're</span> <span m="3193480">trying</span> <span m="3193690">to</span> <span m="3193750">predict</span> <span m="3194050">the</span> <span m="3194140">weather</span> <span m="3194430">for</span> <span m="3194570">Seattle.</span> <span m="3195750">Sometimes</span> <span m="3196280">it</span> <span m="3196490">seems</span> <span m="3196700">like</span> <span m="3196840">this</span> <span m="3196990">in</span> <span m="3197080">Boston.</span> <span m="3198160">And</span> <span m="3198330">you</span> <span m="3198420">just</span> <span m="3198630">say,</span> <span m="3199150">it's</span> <span m="3199620">going</span> <span m="3199730">to</span> <span m="3199820">rain.</span> </p>
<p><span m="3202490">Forget</span> <span m="3202810">all</span> <span m="3202960">the</span> <span m="3203030">fancy</span> <span m="3203410">weather</span> <span m="3203710">forecasting</span> <span m="3204360">stuff,</span> <span m="3204680">the</span> <span m="3204790">radar,</span> <span m="3205330">and</span> <span m="3205400">all</span> <span m="3205580">the</span> <span m="3205680">rest.</span> <span m="3205980">Just</span> <span m="3206190">say</span> <span m="3206420">it's</span> <span m="3206590">going</span> <span m="3206690">to</span> <span m="3206790">rain</span> <span m="3207030">tomorrow.</span> <span m="3208440">You're</span> <span m="3208590">going</span> <span m="3208680">to</span> <span m="3208720">be</span> <span m="3208850">right</span> <span m="3209180">almost</span> <span m="3209510">all</span> <span m="3209660">the</span> <span m="3209750">time.</span> <span m="3210942">All</span> <span m="3211360">right?</span> <span m="3211620">And in</span> <span m="3211720">fact,</span> <span m="3212000">if</span> <span m="3212050">you</span> <span m="3212150">try</span> <span m="3212410">to</span> <span m="3212480">do</span> <span m="3212590">fancy</span> <span m="3213030">stuff,</span> <span m="3214350">you're</span> <span m="3214470">probably</span> <span m="3214760">going</span> <span m="3214870">to</span> <span m="3214930">be</span> <span m="3215620">wrong</span> <span m="3215930">more</span> <span m="3216150">of the</span> <span m="3216270">time.</span> </p>
<p><span m="3217220">All</span> <span m="3217590">right.</span> <span m="3217800">For</span> <span m="3217870">example,</span> <span m="3218370">in</span> <span m="3218470">this</span> <span m="3218490">case,</span> <span m="3219790">if</span> <span m="3219810">you</span> <span m="3219920">just</span> <span m="3220130">say</span> <span m="3220460">the</span> <span m="3220580">person</span> <span m="3221080">does</span> <span m="3221270">not</span> <span m="3221660">have</span> <span m="3221930">the</span> <span m="3222000">disease,</span> <span m="3222820">forget</span> <span m="3223150">the</span> <span m="3223320">lab</span> <span m="3223640">test.</span> <span m="3224510">Just</span> <span m="3224710">come</span> <span m="3224850">back</span> <span m="3225080">with</span> <span m="3225330">negative.</span> <span m="3227620">How</span> <span m="3227850">often</span> <span m="3228090">are</span> <span m="3228210">you right?</span> <span m="3231870">90%</span> <span m="3232250">of</span> <span m="3232520">the</span> <span m="3232620">time</span> <span m="3232950">you're</span> <span m="3233090">right.</span> <span m="3234190">Much</span> <span m="3234520">better</span> <span m="3234800">than</span> <span m="3234950">the</span> <span m="3235040">test</span> <span m="3235390">you</span> <span m="3235480">paid</span> <span m="3235670">a</span> <span m="3235730">lot</span> <span m="3235890">of</span> <span m="3235950">money</span> <span m="3236180">for.</span> </p>
<p><span m="3239050">I</span> <span m="3239270">see.</span> <span m="3239850">You've</span> <span m="3240000">got</span> <span m="3240140">to</span> <span m="3240190">be</span> <span m="3240310">careful</span> <span m="3240890">what</span> <span m="3241020">you're</span> <span m="3241160">looking</span> <span m="3241450">for,</span> <span m="3242630">how</span> <span m="3242840">you</span> <span m="3243020">measure</span> <span m="3243410">the</span> <span m="3243520">value</span> <span m="3243920">of</span> <span m="3243990">a</span> <span m="3244060">test</span> <span m="3244800">or</span> <span m="3245000">a</span> <span m="3245040">prediction.</span> <span m="3246500">Because</span> <span m="3247010">presumably</span> <span m="3247560">the</span> <span m="3247670">one</span> <span m="3247820">you</span> <span m="3247910">paid</span> <span m="3248140">for</span> <span m="3248340">is</span> <span m="3248440">better,</span> <span m="3249820">even</span> <span m="3249940">though</span> <span m="3250740">accurate</span> <span m="3251540">less</span> <span m="3251750">of the</span> <span m="3251870">time.</span> <span m="3254240">Any</span> <span m="3254350">questions</span> <span m="3255040">about</span> <span m="3255522">that?</span> </p>
<p><span m="3260350">OK.</span> <span m="3262390">So</span> <span m="3262670">For</span> <span m="3262720">the</span> <span m="3262800">rest</span> <span m="3263030">of</span> <span m="3263080">today</span> <span m="3263330">we're</span> <span m="3263440">going</span> <span m="3263530">to</span> <span m="3263610">do</span> <span m="3263910">three</span> <span m="3264260">more</span> <span m="3264500">paradoxes.</span> <span m="3267160">And</span> <span m="3267400">in</span> <span m="3267700">each</span> <span m="3267970">case</span> <span m="3268910">they're</span> <span m="3269040">going</span> <span m="3269150">to</span> <span m="3269440">expose</span> <span m="3269990">a</span> <span m="3270070">flaw</span> <span m="3270520">in</span> <span m="3270580">our</span> <span m="3270780">intuition</span> <span m="3271340">about</span> <span m="3271620">probability.</span> <span m="3273220">But</span> <span m="3273680">the</span> <span m="3273800">good</span> <span m="3273970">news</span> <span m="3274170">is in</span> <span m="3274330">each</span> <span m="3274530">case</span> <span m="3274810">it's</span> <span m="3274940">easy</span> <span m="3275260">to</span> <span m="3275310">get</span> <span m="3275450">the</span> <span m="3275520">right</span> <span m="3275690">answer.</span> <span m="3276040">Just</span> <span m="3276570">stick</span> <span m="3276890">with</span> <span m="3277000">the</span> <span m="3277070">math</span> <span m="3277380">and</span> <span m="3277670">try</span> <span m="3277810">not</span> <span m="3277980">to</span> <span m="3278060">think</span> <span m="3278250">about</span> <span m="3278530">it.</span> <span m="3279640">Now</span> <span m="3280000">the</span> <span m="3280260">first</span> <span m="3280600">example</span> <span m="3281200">is</span> <span m="3281410">a</span> <span m="3282200">game</span> <span m="3283650">involving</span> <span m="3284090">dice</span> <span m="3284550">that's</span> <span m="3284700">called</span> <span m="3285200">carnival</span> <span m="3285680">dice</span> <span m="3286710">that</span> <span m="3286850">you</span> <span m="3286980">can</span> <span m="3287280">find in</span> <span m="3287630">carnivals</span> <span m="3288240">and</span> <span m="3288360">you</span> <span m="3288440">can</span> <span m="3288550">also</span> <span m="3288790">find</span> <span m="3289640">in</span> <span m="3289780">casinos.</span> </p>
<p><span m="3293150">It's</span> <span m="3293480">a</span> <span m="3293530">pretty</span> <span m="3293770">popular</span> <span m="3294200">game,</span> <span m="3294680">actually.</span> <span m="3297050">So</span> <span m="3297190">the</span> <span m="3297280">way</span> <span m="3297430">it</span> <span m="3297520">works</span> <span m="3299410">is as</span> <span m="3299470">follows.</span> <span m="3307310">The</span> <span m="3307550">player</span> <span m="3310710">picks</span> <span m="3310970">a</span> <span m="3311310">number</span> <span m="3311680">from</span> <span m="3311880">1</span> <span m="3312090">to</span> <span m="3312170">6--</span> <span m="3314486">we'll</span> <span m="3314930">call</span> <span m="3315290">it N--</span> <span m="3321360">and</span> <span m="3321610">then</span> <span m="3321950">rolls</span> <span m="3322300">three</span> <span m="3322610">dice.</span> <span m="3323650">And</span> <span m="3324140">let's</span> <span m="3324400">say</span> <span m="3324530">they're</span> <span m="3324690">fair</span> <span m="3325210">and</span> <span m="3325270">mutually</span> <span m="3325670">independent.</span> </p>
<p><span m="3335240">We</span> <span m="3335340">haven't</span> <span m="3335550">talked</span> <span m="3335730">about</span> <span m="3335850">independent.</span> <span m="3336470">So</span> <span m="3336640">they're</span> <span m="3336790">fair</span> <span m="3337040">dice.</span> <span m="3338460">For</span> <span m="3338590">now,</span> <span m="3339040">normal</span> <span m="3339400">dice--</span> <span m="3339780">nothing</span> <span m="3340010">fishy.</span> <span m="3341250">And</span> <span m="3341770">the</span> <span m="3341850">player</span> <span m="3343840">wins</span> <span m="3346750">if</span> <span m="3346990">and</span> <span m="3347110">only</span> <span m="3347480">if</span> <span m="3349580">the</span> <span m="3349670">number</span> <span m="3349990">he picked</span> <span m="3350780">comes</span> <span m="3351070">up</span> <span m="3351230">on at</span> <span m="3351480">least</span> <span m="3351820">one</span> <span m="3351990">of</span> <span m="3352040">the</span> <span m="3352150">dice.</span> <span m="3358100">So</span> <span m="3358260">you</span> <span m="3358420">either</span> <span m="3358590">win</span> <span m="3358830">or</span> <span m="3358860">you</span> <span m="3359050">lose</span> <span m="3359420">the</span> <span m="3359500">game</span> <span m="3359830">depending</span> <span m="3360290">on</span> <span m="3360500">if</span> <span m="3360620">your</span> <span m="3361000">lucky</span> <span m="3361340">number</span> <span m="3362300">came</span> <span m="3362580">up</span> <span m="3362720">at</span> <span m="3362800">least</span> <span m="3363080">once.</span> </p>
<p><span m="3365960">Now</span> <span m="3366110">you've</span> <span m="3366250">got</span> <span m="3366510">three</span> <span m="3366800">dice,</span> <span m="3367560">each</span> <span m="3367800">of</span> <span m="3367890">which</span> <span m="3368100">has</span> <span m="3368240">a</span> <span m="3368320">1</span> <span m="3368590">in</span> <span m="3368680">6</span> <span m="3369040">chance</span> <span m="3369570">of</span> <span m="3369670">coming</span> <span m="3369940">up</span> <span m="3370060">a</span> <span m="3370120">winner</span> <span m="3370410">for</span> <span m="3370520">you.</span> <span m="3373420">So</span> <span m="3374700">how</span> <span m="3374870">many</span> <span m="3375080">people</span> <span m="3375800">think</span> <span m="3376040">this</span> <span m="3376230">is</span> <span m="3376340">a</span> <span m="3376410">fair</span> <span m="3376780">game--</span> <span m="3377630">you</span> <span m="3377730">got</span> <span m="3377890">a</span> <span m="3377950">50-50</span> <span m="3378750">chance</span> <span m="3379070">of</span> <span m="3379150">winning--</span> <span m="3380930">three</span> <span m="3381190">dice,</span> <span m="3381690">each</span> <span m="3381910">1/6</span> <span m="3382730">chance</span> <span m="3383030">of</span> <span m="3383110">winning?</span> <span m="3384290">Anybody</span> <span m="3384620">think</span> <span m="3384770">it's</span> <span m="3384920">not</span> <span m="3385170">a</span> <span m="3385220">fair</span> <span m="3385440">game?</span> <span m="3387540">A</span> <span m="3387570">bunch</span> <span m="3387830">of</span> <span m="3387890">you.</span> <span m="3388420">How</span> <span m="3388520">many</span> <span m="3388660">people</span> <span m="3388890">think</span> <span m="3389040">it</span> <span m="3389120">is</span> <span m="3389300">a</span> <span m="3389360">fair</span> <span m="3389590">game--</span> <span m="3390410">50-50?</span> <span m="3391840">A few.</span> <span m="3392830">All right.</span> </p>
<p><span m="3393100">Well,</span> <span m="3393940">let's</span> <span m="3394970">figure</span> <span m="3395290">it</span> <span m="3395340">out.</span> <span m="3396500">And</span> <span m="3396750">instead</span> <span m="3397130">of</span> <span m="3397230">doing</span> <span m="3397460">the</span> <span m="3397550">tree</span> <span m="3397780">method,</span> <span m="3398090">which</span> <span m="3398270">we</span> <span m="3398390">know</span> <span m="3398630">we're</span> <span m="3398740">supposed</span> <span m="3399140">to</span> <span m="3399220">do,</span> <span m="3399530">we're</span> <span m="3399570">just</span> <span m="3399790">going</span> <span m="3399900">to</span> <span m="3400230">wing</span> <span m="3401040">it,</span> <span m="3403860">which</span> <span m="3404240">is</span> <span m="3404380">always</span> <span m="3404870">seems</span> <span m="3405240">easier</span> <span m="3405310">to</span> <span m="3405797">do.</span> <span m="3411650">If</span> <span m="3411930">you're</span> <span m="3412070">in</span> <span m="3412150">the</span> <span m="3412240">Casino</span> <span m="3412790">you</span> <span m="3413300">want</span> <span m="3413450">to</span> <span m="3413500">just</span> <span m="3413790">wing</span> <span m="3414100">it</span> <span m="3414180">instead</span> <span m="3414430">of</span> <span m="3414650">taking</span> <span m="3414960">your</span> <span m="3415000">napkin</span> <span m="3415350">out</span> <span m="3415500">and</span> <span m="3415570">drawing</span> <span m="3415820">a</span> <span m="3415900">tree.</span> </p>
<p><span m="3417970">So</span> <span m="3418230">the</span> <span m="3418340">claim,</span> <span m="3420990">question</span> <span m="3421360">mark,</span> <span m="3421730">is</span> <span m="3422110">the</span> <span m="3422210">probability</span> <span m="3422880">you</span> <span m="3423000">win</span> <span m="3425330">in</span> <span m="3425530">1/2.</span> <span m="3427740">And</span> <span m="3428790">the</span> <span m="3428900">proof,</span> <span m="3429950">question</span> <span m="3430340">mark,</span> <span m="3431840">is</span> <span m="3432150">you</span> <span m="3432320">let</span> <span m="3433690">Ai</span> <span m="3435250">be</span> <span m="3435430">the</span> <span m="3435590">event</span> <span m="3438882">that</span> <span m="3439320">the</span> <span m="3439390">i-th</span> <span m="3439550">die</span> <span m="3441140">comes</span> <span m="3441470">up</span> <span m="3441630">N.</span> <span m="3444160">And</span> <span m="3444300">i</span> <span m="3444460">is</span> <span m="3444600">1</span> <span m="3444850">to</span> <span m="3444920">3</span> <span m="3445220">here.</span> </p>
<p><span m="3446870">So</span> <span m="3447020">then</span> <span m="3447170">you</span> <span m="3447250">say,</span> <span m="3447470">OK.</span> <span m="3447700">The</span> <span m="3447820">probability</span> <span m="3448490">I</span> <span m="3448570">win</span> <span m="3450690">is</span> <span m="3451000">the</span> <span m="3451100">probability</span> <span m="3451920">of</span> <span m="3452560">A1--</span> <span m="3453300">I</span> <span m="3453380">could</span> <span m="3453570">win</span> <span m="3453710">that</span> <span m="3453940">way--</span> <span m="3454220">or</span> <span m="3454550">A2,</span> <span m="3455635">or</span> <span m="3456030">A3.</span> <span m="3458010">All</span> <span m="3458220">I</span> <span m="3458290">need</span> <span m="3458450">is</span> <span m="3458570">one</span> <span m="3458820">of</span> <span m="3458860">the</span> <span m="3458950">die</span> <span m="3459260">to come</span> <span m="3459470">up</span> <span m="3460090">my</span> <span m="3460290">way.</span> <span m="3461810">And</span> <span m="3462190">that</span> <span m="3462560">is</span> <span m="3462940">the</span> <span m="3463020">probability</span> <span m="3463750">of</span> <span m="3463840">A1</span> <span m="3465430">plus</span> <span m="3465770">the</span> <span m="3465860">probability</span> <span m="3466530">of</span> <span m="3466620">A2</span> <span m="3468420">plus</span> <span m="3468750">the</span> <span m="3468860">probability</span> <span m="3469510">of</span> <span m="3469600">A3.</span> <span m="3472210">And</span> <span m="3472440">each</span> <span m="3472730">die</span> <span m="3473010">wins</span> <span m="3473410">for</span> <span m="3473520">me</span> <span m="3473660">with</span> <span m="3473790">probability</span> <span m="3474270">1/6.</span> <span m="3479360">And</span> <span m="3479500">that</span> <span m="3479690">is</span> <span m="3479970">then</span> <span m="3480190">1/2.</span> </p>
<p><span m="3483460">So</span> <span m="3483600">that's</span> <span m="3483810">a</span> <span m="3483870">proof</span> <span m="3484340">that</span> <span m="3485260">we</span> <span m="3485400">win</span> <span m="3485650">with</span> <span m="3485770">probability</span> <span m="3486262">of</span> <span m="3486754">1/2.</span> <span m="3489706">What do you</span> <span m="3490198">think?</span> <span m="3493160">Any</span> <span m="3493270">problems</span> <span m="3493690">with that</span> <span m="3493850">proof?</span> </p>
<p><span m="3494766">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3500730">PROFESSOR: Well</span> <span m="3500880">that's</span> <span m="3501100">a</span> <span m="3501160">great</span> <span m="3501500">point.</span> <span m="3502030">Yeah.</span> <span m="3502850">So</span> <span m="3502970">if</span> <span m="3503060">I</span> <span m="3503200">extended</span> <span m="3503770">this</span> <span m="3504250">nice</span> <span m="3504580">proof</span> <span m="3504820">technique</span> <span m="3505950">I</span> <span m="3506050">couldn't</span> <span m="3506340">have</span> <span m="3506530">probability</span> <span m="3506970">of 7/6</span> <span m="3507730">of</span> <span m="3507810">winning</span> <span m="3508120">with</span> <span m="3508260">seven</span> <span m="3508650">die.</span> <span m="3509110">Yeah?</span> </p>
<p><span m="3510250">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3515100">PROFESSOR: Yeah.</span> <span m="3515540">You're</span> <span m="3516060">very</span> <span m="3516490">close.</span> <span m="3516990">I</span> <span m="3517090">didn't</span> <span m="3518110">technically</span> <span m="3518980">assume</span> <span m="3519295">that.</span> </p>
<p><span m="3521062">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3523970">PROFESSOR: They</span> <span m="3524480">could</span> <span m="3524730">double</span> <span m="3525100">up.</span> <span m="3525450">Yeah.</span> <span m="3526580">There's</span> <span m="3526820">no</span> <span m="3527260">intersection</span> <span m="3528220">in</span> <span m="3528410">the</span> <span m="3528550">events.</span> <span m="3530040">In</span> <span m="3530070">fact,</span> <span m="3531030">there</span> <span m="3531270">is</span> <span m="3531450">intersection</span> <span m="3532080">because</span> <span m="3533010">there's</span> <span m="3533220">a</span> <span m="3533320">chance</span> <span m="3533940">I</span> <span m="3534050">rolled</span> <span m="3534650">all</span> <span m="3534870">six--</span> <span m="3535190">all</span> <span m="3535470">Ns.</span> <span m="3536740">Say</span> <span m="3536900">N</span> <span m="3537210">is</span> <span m="3537430">6.</span> <span m="3537680">I could roll all</span> <span m="3537950">sixes</span> <span m="3538920">and</span> <span m="3539060">then</span> <span m="3539280">each</span> <span m="3539530">of</span> <span m="3539610">these</span> <span m="3539810">would</span> <span m="3539910">be</span> <span m="3540060">a</span> <span m="3540120">winner.</span> <span m="3540990">But</span> <span m="3541140">I</span> <span m="3541210">don't</span> <span m="3541480">get</span> <span m="3541620">to</span> <span m="3541700">count</span> <span m="3541950">them</span> <span m="3543160">separately.</span> <span m="3543640">Then</span> <span m="3543860">I only</span> <span m="3544010">win</span> <span m="3544200">once</span> <span m="3544550">in</span> <span m="3544640">that</span> <span m="3544730">case.</span> </p>
<p><span m="3545020">In</span> <span m="3545070">other</span> <span m="3545230">words,</span> <span m="3546130">all</span> <span m="3546500">of</span> <span m="3546560">these</span> <span m="3546760">could</span> <span m="3546890">turned</span> <span m="3547120">on</span> <span m="3547290">at</span> <span m="3547370">the</span> <span m="3547440">same</span> <span m="3547740">time.</span> <span m="3548060">There's</span> <span m="3548230">an</span> <span m="3548280">intersection</span> <span m="3548880">here.</span> <span m="3549030">So</span> <span m="3549150">this</span> <span m="3549560">rule</span> <span m="3551490">does</span> <span m="3551660">not</span> <span m="3552030">hold.</span> <span m="3553570">I</span> <span m="3553700">need</span> <span m="3554150">the</span> <span m="3554790">Ai</span> <span m="3555210">to</span> <span m="3555310">be</span> <span m="3555410">disjoined</span> <span m="3556070">for</span> <span m="3556140">this</span> <span m="3556290">to</span> <span m="3556380">be</span> <span m="3556460">true--</span> <span m="3557552">the</span> <span m="3557920">events to be</span> <span m="3558360">disjoined.</span> </p>
<p><span m="3560380">And</span> <span m="3560530">they're</span> <span m="3560650">not</span> <span m="3560960">disjoined</span> <span m="3561760">because</span> <span m="3561940">there's</span> <span m="3562110">a</span> <span m="3562190">sample</span> <span m="3562670">point</span> <span m="3563730">were</span> <span m="3564020">two</span> <span m="3564330">or</span> <span m="3564380">more</span> <span m="3564880">of</span> <span m="3564950">the die</span> <span m="3565040">could</span> <span m="3565460">come</span> <span m="3565610">up</span> <span m="3565740">the</span> <span m="3565820">same</span> <span m="3566880">being</span> <span m="3567110">a</span> <span m="3567170">winner,</span> <span m="3568260">which</span> <span m="3568470">means</span> <span m="3569880">the</span> <span m="3570200">same</span> <span m="3570640">sample</span> <span m="3571080">point,</span> <span m="3571420">namely</span> <span m="3571750">all</span> <span m="3572010">die</span> <span m="3572300">are N,</span> <span m="3573560">comes</span> <span m="3573840">up</span> <span m="3573980">in</span> <span m="3574080">each</span> <span m="3574350">of</span> <span m="3574410">these</span> <span m="3574630">three.</span> <span m="3574960">So</span> <span m="3575000">they're</span> <span m="3575140">not</span> <span m="3575490">disjoined.</span> </p>
<p><span m="3576530">Now</span> <span m="3576730">what's</span> <span m="3577050">the</span> <span m="3577340">principal</span> <span m="3577960">you</span> <span m="3578070">used</span> <span m="3579270">two</span> <span m="3579470">weeks</span> <span m="3579730">ago</span> <span m="3580510">when</span> <span m="3580690">you</span> <span m="3580780">did</span> <span m="3581180">cardinality</span> <span m="3581750">of</span> <span m="3581820">a</span> <span m="3581870">set--</span> <span m="3583680">cardinality</span> <span m="3583780">of</span> <span m="3583970">a</span> <span m="3584320">union</span> <span m="3584730">of</span> <span m="3584800">sets?</span> <span m="3586560">Inclusion,</span> <span m="3587290">exclusion.</span> <span m="3588480">And</span> <span m="3588700">the</span> <span m="3588770">same</span> <span m="3589220">thing</span> <span m="3589620">needs</span> <span m="3589760">to</span> <span m="3590230">be done here.</span> </p>
<p><span m="3595870">So</span> <span m="3596060">let's</span> <span m="3596300">do</span> <span m="3596440">that.</span> <span m="3597840">And</span> <span m="3597880">then</span> <span m="3598010">we'll</span> <span m="3598120">figure</span> <span m="3598440">out</span> <span m="3598590">the</span> <span m="3598710">actual</span> <span m="3599020">probability.</span> <span m="3603450">So</span> <span m="3603660">this</span> <span m="3603950">is</span> <span m="3604220">a</span> <span m="3604290">fact</span> <span m="3604750">based</span> <span m="3605010">on</span> <span m="3605090">the</span> <span m="3605210">inclusion,</span> <span m="3605650">exclusion</span> <span m="3606150">principle.</span> <span m="3608130">The</span> <span m="3608230">probability</span> <span m="3608800">of</span> <span m="3609290">A1,</span> <span m="3609860">union</span> <span m="3610180">A2,</span> <span m="3611470">union</span> <span m="3611610">A3,</span> <span m="3613510">is</span> <span m="3613730">just</span> <span m="3614040">what</span> <span m="3614160">you</span> <span m="3614280">think</span> <span m="3614500">it</span> <span m="3614590">would</span> <span m="3614710">be</span> <span m="3615300">from</span> <span m="3615890">inclusion,</span> <span m="3616340">exclusion.</span> </p>
<p><span m="3617680">It's</span> <span m="3617780">a</span> <span m="3617830">probability</span> <span m="3618280">of A1</span> <span m="3618810">plus</span> <span m="3619070">a</span> <span m="3619150">probability</span> <span m="3619610">of</span> <span m="3619700">A2</span> <span m="3621160">plus</span> <span m="3621450">the</span> <span m="3621530">probability</span> <span m="3622050">of</span> <span m="3622140">A3</span> <span m="3624720">minus</span> <span m="3626550">the</span> <span m="3626650">pairwise</span> <span m="3627330">intersections.</span> <span m="3633700">A1</span> <span m="3634130">intersect</span> <span m="3634445">A3</span> <span m="3636650">minus</span> <span m="3637775">probability</span> <span m="3638030">of</span> <span m="3638310">A2</span> <span m="3639200">intersect</span> <span m="3639660">A3.</span> <span m="3642310">And</span> <span m="3642570">is</span> <span m="3642700">there</span> <span m="3643040">anything</span> <span m="3643350">else?</span> <span m="3645860">Plus,</span> <span m="3646400">the</span> <span m="3646490">probably</span> <span m="3646810">of</span> <span m="3646890">all</span> <span m="3647360">of</span> <span m="3647410">them</span> <span m="3647870">matching.</span> </p>
<p><span m="3654930">OK.</span> <span m="3655360">So</span> <span m="3656180">the</span> <span m="3656310">proof</span> <span m="3657960">is</span> <span m="3658150">really</span> <span m="3658370">the</span> <span m="3658460">same</span> <span m="3658780">proof</span> <span m="3659070">you</span> <span m="3659140">use</span> <span m="3659410">for</span> <span m="3659520">inclusion,</span> <span m="3659930">exclusion</span> <span m="3660700">with</span> <span m="3660920">sets.</span> <span m="3661720">The</span> <span m="3661870">only</span> <span m="3662080">difference</span> <span m="3662500">is</span> <span m="3662720">that</span> <span m="3662840">in a</span> <span m="3662900">probability</span> <span m="3663420">space,</span> <span m="3663800">we</span> <span m="3663920">have</span> <span m="3664090">weights</span> <span m="3664680">on</span> <span m="3664850">the</span> <span m="3664960">elements.</span> <span m="3665710">And</span> <span m="3665840">the</span> <span m="3665940">weight</span> <span m="3666540">corresponds</span> <span m="3667100">to</span> <span m="3667200">the</span> <span m="3667300">probability.</span> </p>
<p><span m="3669190">So</span> <span m="3669480">in</span> <span m="3669560">fact,</span> <span m="3669950">if</span> <span m="3670050">you</span> <span m="3670150">were</span> <span m="3670250">drawing</span> <span m="3670540">the</span> <span m="3670630">sample</span> <span m="3671040">space,</span> <span m="3672150">say</span> <span m="3672380">here's</span> <span m="3672720">A1</span> <span m="3674510">and</span> <span m="3674900">here's</span> <span m="3675230">A2,</span> <span m="3676830">and</span> <span m="3676970">here's</span> <span m="3677400">A3.</span> <span m="3679790">Well,</span> <span m="3680030">you</span> <span m="3680140">need</span> <span m="3680410">to</span> <span m="3681490">add</span> <span m="3681830">the</span> <span m="3681910">probabilities</span> <span m="3682580">here,</span> <span m="3684030">here,</span> <span m="3684860">and</span> <span m="3684900">here.</span> <span m="3685770">Then</span> <span m="3685950">you</span> <span m="3686030">subtract</span> <span m="3686660">off</span> <span m="3686950">the</span> <span m="3687050">double</span> <span m="3687350">counting</span> <span m="3687780">from</span> <span m="3687940">here,</span> <span m="3689200">from</span> <span m="3689310">here,</span> <span m="3690180">and</span> <span m="3690290">from</span> <span m="3690440">here.</span> <span m="3690790">And</span> <span m="3690900">then</span> <span m="3691020">you</span> <span m="3691200">add</span> <span m="3691490">back</span> <span m="3691800">again</span> <span m="3692610">what</span> <span m="3692750">you</span> <span m="3692810">subtracted</span> <span m="3693350">off</span> <span m="3693560">too</span> <span m="3693720">much</span> <span m="3694840">there.</span> <span m="3696460">Same</span> <span m="3696810">proof,</span> <span m="3697160">it's</span> <span m="3697290">just</span> <span m="3697730">your</span> <span m="3698070">have</span> <span m="3698240">weights</span> <span m="3698860">on</span> <span m="3698950">the</span> <span m="3699040">elements</span> <span m="3699430">of</span> <span m="3699530">probabilities.</span> </p>
<p><span m="3702120">All</span> <span m="3702250">right.</span> <span m="3702440">So</span> <span m="3702550">let's</span> <span m="3702720">figure</span> <span m="3702960">out</span> <span m="3703060">the</span> <span m="3703290">right</span> <span m="3703510">probability.</span> <span m="3705280">That's</span> <span m="3705460">1/6,</span> <span m="3706630">1/6,</span> <span m="3708905">1/6.</span> <span m="3710550">What's</span> <span m="3710780">the</span> <span m="3710850">probability</span> <span m="3711450">of</span> <span m="3712790">the</span> <span m="3712910">first</span> <span m="3713270">two</span> <span m="3713430">die</span> <span m="3713700">matching--</span> <span m="3714480">both</span> <span m="3714730">of</span> <span m="3714790">them?</span> <span m="3720360">1/36.</span> <span m="3722640">We'll</span> <span m="3722820">talk</span> <span m="3723100">more</span> <span m="3723280">about</span> <span m="3723490">why</span> <span m="3723690">that</span> <span m="3723900">is</span> <span m="3724670">next</span> <span m="3724920">time.</span> <span m="3725110">But</span> <span m="3725210">there's</span> <span m="3725750">a</span> <span m="3725810">6</span> <span m="3726410">for</span> <span m="3726560">A1</span> <span m="3727460">then</span> <span m="3727640">given</span> <span m="3727820">that</span> <span m="3727990">1/6</span> <span m="3728315">for</span> <span m="3729100">the</span> <span m="3729240">second</span> <span m="3729600">die</span> <span m="3729690">matching.</span> <span m="3730260">So</span> <span m="3730610">it's</span> <span m="3730730">1/6</span> <span m="3731000">times</span> <span m="3731180">1/6</span> <span m="3732880">minus the</span> <span m="3733330">1/36.</span> <span m="3735800">1/36,</span> <span m="3736490">the</span> <span m="3736600">chance</span> <span m="3736970">that</span> <span m="3737130">all</span> <span m="3737490">three</span> <span m="3737830">match</span> <span m="3738910">is</span> <span m="3739060">1/216</span> <span m="3740280">or</span> <span m="3740390">6</span> <span m="3740730">cubed.</span> <span m="3742010">So</span> <span m="3742170">when</span> <span m="3742240">you</span> <span m="3742310">add</span> <span m="3742680">all</span> <span m="3742870">that</span> <span m="3743090">up</span> <span m="3744500">you</span> <span m="3744640">get</span> <span m="3745370">the</span> <span m="3745420">0.421</span> <span m="3747750">and</span> <span m="3747960">some</span> <span m="3748100">more.</span> </p>
<p><span m="3749520">So</span> <span m="3749650">the</span> <span m="3749740">chance</span> <span m="3750140">of</span> <span m="3750240">winning</span> <span m="3750530">this</span> <span m="3750730">game</span> <span m="3751060">is</span> <span m="3751200">41%</span> <span m="3752880">which</span> <span m="3753180">makes</span> <span m="3753450">it</span> <span m="3753590">a</span> <span m="3753880">worst</span> <span m="3754320">game</span> <span m="3754640">in</span> <span m="3754710">the</span> <span m="3754780">casino.</span> <span m="3756280">It</span> <span m="3756410">is</span> <span m="3756540">hard</span> <span m="3757010">to</span> <span m="3757090">find</span> <span m="3757500">a</span> <span m="3757550">worse</span> <span m="3757800">game</span> <span m="3758040">than this.</span> <span m="3758560">Roulette,</span> <span m="3759060">much</span> <span m="3759370">better.</span> <span m="3759580">We'll</span> <span m="3759680">study</span> <span m="3759940">Roulette</span> <span m="3760300">in</span> <span m="3760360">the</span> <span m="3760450">last</span> <span m="3760750">lecture--</span> <span m="3761460">much</span> <span m="3761780">better</span> <span m="3761990">game.</span> <span m="3762960">And</span> <span m="3763070">even</span> <span m="3763230">that's</span> <span m="3763450">a</span> <span m="3763490">terrible</span> <span m="3763850">game to</span> <span m="3764040">play.</span> </p>
<p><span m="3765990">So</span> <span m="3766550">it</span> <span m="3766730">looks</span> <span m="3766970">like</span> <span m="3767100">an</span> <span m="3767190">easy</span> <span m="3767400">game.</span> <span m="3767700">There's</span> <span m="3767870">a</span> <span m="3768100">quick</span> <span m="3768340">proof</span> <span m="3768620">that</span> <span m="3768730">it's</span> <span m="3768950">50-50.</span> <span m="3769600">But</span> <span m="3769750">it's</span> <span m="3769880">horrible</span> <span m="3770390">odds</span> <span m="3771360">against</span> <span m="3771680">the</span> <span m="3771750">house.</span> <span m="3774250">Now,</span> <span m="3775720">this</span> <span m="3775980">is</span> <span m="3776080">a</span> <span m="3776140">nice</span> <span m="3776370">example</span> <span m="3776980">because</span> <span m="3777550">it</span> <span m="3777660">shows</span> <span m="3778080">how</span> <span m="3778910">a</span> <span m="3779110">rule</span> <span m="3779430">you</span> <span m="3779560">had</span> <span m="3780050">for</span> <span m="3780250">computing</span> <span m="3780810">the</span> <span m="3780900">cardinality</span> <span m="3781490">of</span> <span m="3781560">a</span> <span m="3781610">set</span> <span m="3782830">gives</span> <span m="3783120">you</span> <span m="3783570">the</span> <span m="3783680">probability.</span> <span m="3785100">All right.</span> </p>
<p><span m="3785620">In</span> <span m="3785790">fact,</span> <span m="3786130">all</span> <span m="3786470">the</span> <span m="3786580">set</span> <span m="3786930">laws</span> <span m="3788210">you</span> <span m="3788760">learned</span> <span m="3788990">a</span> <span m="3789030">couple</span> <span m="3789260">weeks</span> <span m="3789480">ago</span> <span m="3790210">work</span> <span m="3790720">for</span> <span m="3790920">probability</span> <span m="3791480">spaces</span> <span m="3791950">the</span> <span m="3792060">same</span> <span m="3792470">way.</span> <span m="3793130">And</span> <span m="3793280">there</span> <span m="3793360">were</span> <span m="3793450">several</span> <span m="3793870">of</span> <span m="3793970">those</span> <span m="3794120">in</span> <span m="3794210">homework</span> <span m="3794760">that</span> <span m="3794930">you just</span> <span m="3795150">had</span> <span m="3795310">the</span> <span m="3795390">last</span> <span m="3795630">problem</span> <span m="3795950">set.</span> <span m="3798400">Any</span> <span m="3799010">questions</span> <span m="3799485">about that?</span> </p>
<p><span m="3805660">OK.</span> <span m="3807090">Now</span> <span m="3807330">in</span> <span m="3807480">addition,</span> <span m="3808330">all those</span> <span m="3808800">set</span> <span m="3809180">laws</span> <span m="3809510">you</span> <span m="3809640">did</span> <span m="3811340">also</span> <span m="3811800">work</span> <span m="3812210">for</span> <span m="3812360">conditional</span> <span m="3812910">probabilities.</span> <span m="3815070">For</span> <span m="3815110">example,</span> <span m="3816900">this</span> <span m="3817100">is</span> <span m="3817200">true.</span> <span m="3818410">The</span> <span m="3818460">probability</span> <span m="3818930">of</span> <span m="3819050">A</span> <span m="3819430">union</span> <span m="3819810">B</span> <span m="3820970">given</span> <span m="3821300">C--</span> <span m="3821750">whoops--</span> <span m="3822920">given</span> <span m="3823160">C,</span> <span m="3824690">is</span> <span m="3825170">the</span> <span m="3825260">probability</span> <span m="3826130">of</span> <span m="3826410">A</span> <span m="3827890">given</span> <span m="3828170">C</span> <span m="3829020">plus</span> <span m="3829580">the</span> <span m="3829670">probability</span> <span m="3830380">of</span> <span m="3830460">B</span> <span m="3830860">given</span> <span m="3831160">C</span> <span m="3832530">minus</span> <span m="3833670">the</span> <span m="3833790">intersection,</span> <span m="3834450">A</span> <span m="3835290">intersect</span> <span m="3835810">B</span> <span m="3836770">given</span> <span m="3837030">C.</span> <span m="3838310">In</span> <span m="3838430">other</span> <span m="3838610">words,</span> <span m="3839000">take</span> <span m="3839200">any</span> <span m="3839610">probability</span> <span m="3840180">rule</span> <span m="3840420">you</span> <span m="3840510">have</span> <span m="3842180">and</span> <span m="3842370">condition</span> <span m="3842970">everything</span> <span m="3843490">on an</span> <span m="3843690">event,</span> <span m="3844040">C,</span> <span m="3845690">and it</span> <span m="3845890">still</span> <span m="3846120">works.</span> </p>
<p><span m="3849310">And</span> <span m="3849440">the</span> <span m="3849530">proof</span> <span m="3849790">is</span> <span m="3849910">not</span> <span m="3850120">hard.</span> <span m="3851340">You</span> <span m="3851635">can</span> <span m="3851930">go</span> <span m="3852080">through</span> <span m="3852220">each</span> <span m="3852480">individual</span> <span m="3853010">law</span> <span m="3853580">but</span> <span m="3853710">it</span> <span m="3853920">all</span> <span m="3854190">comes</span> <span m="3854420">out</span> <span m="3854670">to</span> <span m="3854720">be</span> <span m="3854840">fine.</span> <span m="3857550">All</span> <span m="3857690">right.</span> <span m="3859680">You have</span> <span m="3859790">to be a</span> <span m="3859880">little</span> <span m="3860160">careful</span> <span m="3860530">though</span> <span m="3860690">because</span> <span m="3861690">you</span> <span m="3861770">got</span> <span m="3862500">to</span> <span m="3862650">remember</span> <span m="3862880">which</span> <span m="3863140">side</span> <span m="3863550">you're</span> <span m="3863650">doing,</span> <span m="3865030">which</span> <span m="3865320">what</span> <span m="3865490">you're</span> <span m="3865590">putting</span> <span m="3865870">on</span> <span m="3865990">either</span> <span m="3866180">side</span> <span m="3866400">of</span> <span m="3866460">the</span> <span m="3866540">bar</span> <span m="3866840">here.</span> <span m="3867580">For</span> <span m="3867680">example,</span> <span m="3868020">what</span> <span m="3868140">about</span> <span m="3868350">this</span> <span m="3868540">one?</span> <span m="3869830">Is this</span> <span m="3870070">true?</span> </p>
<p><span m="3873080">Claim.</span> <span m="3874900">Let's</span> <span m="3874990">take--</span> <span m="3875950">say</span> <span m="3876230">C</span> <span m="3876590">and</span> <span m="3876700">D</span> <span m="3876860">are</span> <span m="3876910">disjoined.</span> <span m="3882900">Is</span> <span m="3883070">this</span> <span m="3883230">true?</span> <span m="3885690">Then</span> <span m="3885810">the</span> <span m="3885910">probability</span> <span m="3887640">of</span> <span m="3887870">A</span> <span m="3889310">conditioned</span> <span m="3889870">on</span> <span m="3890010">C union</span> <span m="3890860">D.</span> <span m="3893330">So</span> <span m="3893500">given</span> <span m="3893770">that</span> <span m="3893910">either</span> <span m="3894260">C</span> <span m="3894490">or</span> <span m="3894550">D</span> <span m="3894740">is</span> <span m="3894900">true,</span> <span m="3896450">does</span> <span m="3896600">that</span> <span m="3896790">equal</span> <span m="3897030">the</span> <span m="3897140">probability</span> <span m="3897740">of</span> <span m="3897840">A</span> <span m="3899440">given</span> <span m="3899740">C</span> <span m="3900810">plus</span> <span m="3901960">probability</span> <span m="3902470">of</span> <span m="3902560">A</span> <span m="3903820">given</span> <span m="3904140">D?</span> </p>
<p><span m="3907140">We</span> <span m="3907270">know</span> <span m="3907570">that</span> <span m="3907700">if</span> <span m="3907830">I</span> <span m="3908300">swapped</span> <span m="3908940">all</span> <span m="3909120">these,</span> <span m="3909690">it's</span> <span m="3909870">true.</span> <span m="3911150">The</span> <span m="3911260">probability</span> <span m="3911810">of</span> <span m="3911880">C</span> <span m="3912240">union</span> <span m="3912620">D</span> <span m="3912990">when</span> <span m="3913410">C</span> <span m="3913620">and</span> <span m="3913700">D</span> <span m="3913810">are</span> <span m="3913870">disjoined</span> <span m="3914500">is</span> <span m="3914770">the</span> <span m="3914860">probability</span> <span m="3915250">that</span> <span m="3915310">C</span> <span m="3915540">given</span> <span m="3915800">A</span> <span m="3915980">plus</span> <span m="3916250">the probability</span> <span m="3916530">of</span> <span m="3916650">D</span> <span m="3916820">given</span> <span m="3917050">A.</span> <span m="3918180">That</span> <span m="3918410">I</span> <span m="3918470">just</span> <span m="3918680">claimed.</span> </p>
<p><span m="3919030">And</span> <span m="3919100">what</span> <span m="3919300">about</span> <span m="3919570">this</span> <span m="3919750">way?</span> <span m="3920070">Can I</span> <span m="3920640">swap</span> <span m="3921110">things</span> <span m="3921400">around?</span> <span m="3924580">Yeah?</span> </p>
<p><span m="3926306">AUDIENCE: [INAUDIBLE]</span> <span m="3929750">would</span> <span m="3930242">C union</span> <span m="3930734">D be</span> <span m="3931226">0?</span> </p>
<p><span m="3932710">PROFESSOR: If</span> <span m="3933210">C</span> <span m="3933580">and</span> <span m="3933950">D are</span> <span m="3934150">disjoined,</span> <span m="3936350">C</span> <span m="3936810">union</span> <span m="3937310">D</span> <span m="3938320">would</span> <span m="3938450">just</span> <span m="3938640">be</span> <span m="3938740">C</span> <span m="3938980">union</span> <span m="3939300">D.</span> <span m="3942170">But</span> <span m="3942280">you're</span> <span m="3942360">not</span> <span m="3942530">a</span> <span m="3942580">good</span> <span m="3942770">point.</span> <span m="3943050">What</span> <span m="3943210">if</span> <span m="3943290">C</span> <span m="3943580">and</span> <span m="3943700">D</span> <span m="3943860">are</span> <span m="3943930">disjoined?</span> <span m="3944540">That's</span> <span m="3944750">a</span> <span m="3945000">good</span> <span m="3945150">example.</span> <span m="3945918">Let's draw that.</span> <span m="3952750">Let's look at that</span> <span m="3953238">case.</span> </p>
<p><span m="3957640">So</span> <span m="3957760">we've</span> <span m="3957930">got</span> <span m="3958220">a</span> <span m="3958280">sample</span> <span m="3958700">space</span> <span m="3959060">here.</span> <span m="3960540">And</span> <span m="3960790">you've</span> <span m="3960880">got</span> <span m="3961640">C</span> <span m="3962140">here</span> <span m="3963360">and</span> <span m="3963600">D</span> <span m="3963890">here.</span> <span m="3965560">And</span> <span m="3965700">just</span> <span m="3965870">for</span> <span m="3965960">fun,</span> <span m="3966260">let's</span> <span m="3966470">make</span> <span m="3966670">A</span> <span m="3967000">be</span> <span m="3967190">here--</span> <span m="3968660">include</span> <span m="3969040">all</span> <span m="3969260">of them.</span> <span m="3973140">What's</span> <span m="3973400">the</span> <span m="3973510">probability--</span> <span m="3976372">is this going</span> <span m="3976840">to</span> <span m="3977440">do</span> <span m="3977540">what</span> <span m="3977640">I</span> <span m="3977700">want?</span> <span m="3978680">Yeah.</span> <span m="3979230">What's</span> <span m="3979500">the</span> <span m="3979590">probability</span> <span m="3980130">of</span> <span m="3980260">A</span> <span m="3980520">given</span> <span m="3981012">C?</span> <span m="3985440">1.</span> <span m="3986330">If</span> <span m="3986470">I'm in</span> <span m="3986760">C</span> <span m="3987100">I'm</span> <span m="3987220">in</span> <span m="3987350">A.</span> <span m="3987940">A is</span> <span m="3988100">everything</span> <span m="3988470">here.</span> <span m="3990300">So</span> <span m="3990480">the</span> <span m="3990540">probability</span> <span m="3990970">of A</span> <span m="3991190">given</span> <span m="3991430">C</span> <span m="3991640">is</span> <span m="3991770">one.</span> <span m="3992970">What's</span> <span m="3993210">the</span> <span m="3993300">probability</span> <span m="3993840">of</span> <span m="3994230">A</span> <span m="3994460">given</span> <span m="3994770">D?</span> <span m="3996255">1.</span> </p>
<p><span m="3999225">All right.</span> <span m="3999720">Well, this is a</span> <span m="3999790">problem</span> <span m="4000970">because</span> <span m="4001240">I</span> <span m="4001320">can't</span> <span m="4001670">have</span> <span m="4001880">the</span> <span m="4001960">probability</span> <span m="4002490">ot--</span> <span m="4002750">what's</span> <span m="4002930">the</span> <span m="4003000">probably</span> <span m="4003280">of</span> <span m="4003390">A</span> <span m="4003630">given</span> <span m="4003860">C</span> <span m="4004140">union</span> <span m="4004510">D?</span> <span m="4006890">Well,</span> <span m="4007090">it</span> <span m="4007150">can't</span> <span m="4007380">be</span> <span m="4007490">2.</span> <span m="4007790">Right?</span> <span m="4008950">It's</span> <span m="4009140">1.</span> <span m="4010550">They</span> <span m="4010740">are</span> <span m="4010810">not</span> <span m="4011070">equal.</span> </p>
<p><span m="4014120">So</span> <span m="4014380">you</span> <span m="4014580">cannot</span> <span m="4016370">do</span> <span m="4016510">those</span> <span m="4016770">set</span> <span m="4017090">rules</span> <span m="4017540">on</span> <span m="4017730">the</span> <span m="4017860">right</span> <span m="4018320">side</span> <span m="4018660">of</span> <span m="4018710">the</span> <span m="4018800">conditioning</span> <span m="4019340">bar.</span> <span m="4021020">You</span> <span m="4021180">can</span> <span m="4021320">do</span> <span m="4021420">them</span> <span m="4021600">on</span> <span m="4021740">the</span> <span m="4021830">left,</span> <span m="4022560">not</span> <span m="4022970">on</span> <span m="4023120">the</span> <span m="4023210">right.</span> <span m="4024220">All right.</span> <span m="4024580">So this</span> <span m="4024830">is not true.</span> <span m="4033320">Now</span> <span m="4033440">nobody</span> <span m="4033800">would</span> <span m="4034000">do</span> <span m="4034160">this.</span> <span m="4034430">Right?</span> <span m="4034890">I</span> <span m="4034960">mean,</span> <span m="4035060">the</span> <span m="4035130">probability</span> <span m="4035780">of--</span> <span m="4036700">not</span> <span m="4036850">that</span> <span m="4037070">it's--</span> <span m="4037370">see</span> <span m="4037500">this</span> <span m="4037690">example?</span> </p>
<p><span m="4038140">This</span> <span m="4038940">you</span> <span m="4039070">just</span> <span m="4039280">would</span> <span m="4039420">never</span> <span m="4039780">make</span> <span m="4040020">this</span> <span m="4040200">mistake</span> <span m="4040550">again</span> <span m="4041270">seeing</span> <span m="4041520">that</span> <span m="4041670">example.</span> <span m="4043450">Everybody</span> <span m="4043770">understand</span> <span m="4044150">the example,</span> <span m="4045420">how</span> <span m="4045560">it's</span> <span m="4045710">clearly</span> <span m="4046110">not</span> <span m="4046430">always</span> <span m="4046720">the</span> <span m="4046820">case</span> <span m="4047170">that</span> <span m="4047440">probability</span> <span m="4047790">of</span> <span m="4047870">A</span> <span m="4048010">given</span> <span m="4048290">C</span> <span m="4048510">union</span> <span m="4048770">D</span> <span m="4048940">is</span> <span m="4049060">a</span> <span m="4049270">probability</span> <span m="4049410">of</span> <span m="4049510">A</span> <span m="4050130">given</span> <span m="4050320">C</span> <span m="4050530">plus</span> <span m="4050770">probability</span> <span m="4051050">of</span> <span m="4051150">A</span> <span m="4051270">given</span> <span m="4051500">D?</span> </p>
<p><span m="4053180">Because</span> <span m="4053330">now</span> <span m="4053440">I'm</span> <span m="4053560">going</span> <span m="4053640">to</span> <span m="4053690">show</span> <span m="4053910">you</span> <span m="4054050">an</span> <span m="4054130">example</span> <span m="4055140">where you're</span> <span m="4055390">going</span> <span m="4055480">to</span> <span m="4055540">swear</span> <span m="4055830">it's</span> <span m="4055930">true.</span> <span m="4058250">All</span> <span m="4058330">right?</span> <span m="4059440">And</span> <span m="4059560">this</span> <span m="4059690">is</span> <span m="4059790">a</span> <span m="4059860">real</span> <span m="4060040">life</span> <span m="4060270">example.</span> <span m="4063660">Many</span> <span m="4063970">years</span> <span m="4064240">ago</span> <span m="4064430">now</span> <span m="4065320">there</span> <span m="4065470">was</span> <span m="4065630">a</span> <span m="4065730">sex</span> <span m="4066040">discrimination</span> <span m="4066730">suit</span> <span m="4067270">at</span> <span m="4067430">Berkeley.</span> </p>
<p><span m="4069320">There</span> <span m="4069510">was</span> <span m="4069670">a</span> <span m="4069770">female</span> <span m="4070140">professor</span> <span m="4070990">in</span> <span m="4071140">the</span> <span m="4071230">math</span> <span m="4071590">department.</span> <span m="4072170">And</span> <span m="4072310">she</span> <span m="4072440">was</span> <span m="4072670">denied</span> <span m="4073010">tenure.</span> <span m="4074500">And</span> <span m="4074910">she</span> <span m="4075710">filed</span> <span m="4075960">a</span> <span m="4076010">lawsuit</span> <span m="4076500">against</span> <span m="4076810">Berkeley</span> <span m="4077180">alleging</span> <span m="4077830">sex</span> <span m="4078120">discrimination.</span> <span m="4079680">Said</span> <span m="4079750">she</span> <span m="4079880">wasn't</span> <span m="4080250">tenured</span> <span m="4080550">because</span> <span m="4080800">she's</span> <span m="4080980">a</span> <span m="4081050">woman.</span> </p>
<p><span m="4082390">Now,</span> <span m="4082840">unfortunately</span> <span m="4083850">sex</span> <span m="4084190">discrimination</span> <span m="4084810">is</span> <span m="4085040">a</span> <span m="4085110">problem</span> <span m="4085420">in</span> <span m="4085490">math</span> <span m="4085740">departments.</span> <span m="4086035">It's</span> <span m="4086330">historically</span> <span m="4086930">been</span> <span m="4087110">a</span> <span m="4087950">difficult</span> <span m="4088430">area.</span> <span m="4089900">But</span> <span m="4090120">it's</span> <span m="4090250">always</span> <span m="4090540">hard</span> <span m="4090820">to</span> <span m="4090900">prove.</span> <span m="4091340">It's</span> <span m="4091430">a</span> <span m="4091980">nebulous</span> <span m="4092570">kind</span> <span m="4092740">of</span> <span m="4092840">thing.</span> </p>
<p><span m="4093120">They</span> <span m="4093200">don't</span> <span m="4093420">say,</span> <span m="4093730">hey,</span> <span m="4093920">you</span> <span m="4094050">can't</span> <span m="4094260">have</span> <span m="4094370">tenure</span> <span m="4094590">because</span> <span m="4094760">you're</span> <span m="4094980">a</span> <span m="4095040">woman.</span> <span m="4095810">They'd</span> <span m="4095870">get</span> <span m="4096029">sued</span> <span m="4096520">and</span> <span m="4096750">get</span> <span m="4097140">killed</span> <span m="4097520">for</span> <span m="4097609">that.</span> <span m="4099550">So</span> <span m="4100000">she</span> <span m="4100180">had</span> <span m="4100390">to</span> <span m="4100490">get</span> <span m="4100910">some</span> <span m="4101420">mat</span> <span m="4101740">to back</span> <span m="4102229">her up.</span> <span m="4104020">So</span> <span m="4104220">what</span> <span m="4104350">she</span> <span m="4104460">did</span> <span m="4104700">is</span> <span m="4104770">she</span> <span m="4105359">looked</span> <span m="4105609">into</span> <span m="4106319">Berkeley's</span> <span m="4106890">practices</span> <span m="4107569">and</span> <span m="4107670">she</span> <span m="4107810">found</span> <span m="4108279">that</span> <span m="4108370">in</span> <span m="4108689">all</span> <span m="4109149">22</span> <span m="4109779">departments,</span> <span m="4111000">every</span> <span m="4111290">single</span> <span m="4111580">department,</span> <span m="4113220">the</span> <span m="4113350">percentage</span> <span m="4114040">of</span> <span m="4114149">male</span> <span m="4114910">PhD</span> <span m="4115510">applicants</span> <span m="4116670">that</span> <span m="4116800">were</span> <span m="4116960">accepted</span> <span m="4118430">was</span> <span m="4118640">higher</span> <span m="4120069">than</span> <span m="4120210">the</span> <span m="4120310">percentage</span> <span m="4121540">of</span> <span m="4121670">female</span> <span m="4122390">PhD</span> <span m="4123380">applicants</span> <span m="4123950">that</span> <span m="4124060">were</span> <span m="4124200">accepted.</span> </p>
<p><span m="4126069">Now</span> <span m="4126160">you</span> <span m="4126300">could</span> <span m="4126430">understand</span> <span m="4127210">some</span> <span m="4127399">of</span> <span m="4127439">the</span> <span m="4127500">departments</span> <span m="4128000">accepting</span> <span m="4128430">more</span> <span m="4129069">male</span> <span m="4129340">PhDs</span> <span m="4129970">than</span> <span m="4130200">female</span> <span m="4130510">PhDs.</span> <span m="4130809">But</span> <span m="4131109">all</span> <span m="4131540">22?</span> <span m="4133430">What</span> <span m="4134040">are</span> <span m="4134060">the</span> <span m="4134210">odds</span> <span m="4134390">of</span> <span m="4134470">that?</span> <span m="4134740">I</span> <span m="4134760">mean,</span> <span m="4134930">so</span> <span m="4135100">the</span> <span m="4135500">immediate</span> <span m="4135880">conclusion</span> <span m="4136450">is,</span> <span m="4137609">well,</span> <span m="4138100">that's</span> <span m="4138310">clearly</span> <span m="4138819">there's</span> <span m="4138960">sex</span> <span m="4139250">discrimination</span> <span m="4139890">going</span> <span m="4140100">on</span> <span m="4140220">at</span> <span m="4140270">Berkeley.</span> <span m="4141500">OK?</span> </p>
<p><span m="4143399">Well</span> <span m="4143710">Berkeley</span> <span m="4144060">took</span> <span m="4144210">a</span> <span m="4144260">look</span> <span m="4144430">at</span> <span m="4144490">that</span> <span m="4144700">and</span> <span m="4144800">said,</span> <span m="4145649">nothing</span> <span m="4145899">good.</span> <span m="4146260">That</span> <span m="4146430">doesn't</span> <span m="4146649">look</span> <span m="4146790">good</span> <span m="4146950">for</span> <span m="4147060">them.</span> <span m="4148130">But</span> <span m="4148260">they</span> <span m="4148359">did</span> <span m="4148529">their</span> <span m="4148700">own</span> <span m="4148899">study</span> <span m="4150520">of</span> <span m="4150660">PhD</span> <span m="4151210">applicants.</span> <span m="4153140">And</span> <span m="4153470">they</span> <span m="4153660">said</span> <span m="4154790">that</span> <span m="4155090">if</span> <span m="4155520">the</span> <span m="4155609">university</span> <span m="4156180">as</span> <span m="4156290">a</span> <span m="4156370">whole--</span> <span m="4156930">look at</span> <span m="4157000">the</span> <span m="4157060">University</span> <span m="4157470">as</span> <span m="4157580">a</span> <span m="4157630">whole,</span> <span m="4158890">actually,</span> <span m="4160490">the</span> <span m="4160609">women,</span> <span m="4161040">the</span> <span m="4161130">females</span> <span m="4162050">have</span> <span m="4162210">a</span> <span m="4162260">higher</span> <span m="4162740">acceptance</span> <span m="4163330">rate</span> <span m="4164359">for</span> <span m="4164450">the</span> <span m="4164560">PhD</span> <span m="4165029">Program</span> <span m="4165840">than</span> <span m="4165990">the</span> <span m="4166060">men.</span> </p>
<p><span m="4167750">So</span> <span m="4167920">look.</span> <span m="4168109">Berkeley</span> <span m="4168200">said,</span> <span m="4168729">we're</span> <span m="4168870">accepting</span> <span m="4169330">more</span> <span m="4169540">women</span> <span m="4169840">than</span> <span m="4169970">men</span> <span m="4170810">percentage-wise.</span> <span m="4172540">So</span> <span m="4172790">how</span> <span m="4173100">could</span> <span m="4173229">we</span> <span m="4173340">be</span> <span m="4173430">discriminating</span> <span m="4174080">against</span> <span m="4174350">women?</span> <span m="4175620">And</span> <span m="4175760">this</span> <span m="4175920">is</span> <span m="4176020">where</span> <span m="4176340">the</span> <span m="4176779">same</span> <span m="4177170">argument</span> <span m="4177609">the</span> <span m="4178020">female</span> <span m="4178350">faculty</span> <span m="4178670">member's</span> <span m="4178960">making,</span> <span m="4179939">But</span> <span m="4180100">they're</span> <span m="4180210">saying</span> <span m="4180420">as</span> <span m="4180520">a</span> <span m="4180580">university</span> <span m="4181010">as</span> <span m="4181130">a</span> <span m="4181200">whole,</span> <span m="4181490">when</span> <span m="4181580">you</span> <span m="4181649">add</span> <span m="4181790">up</span> <span m="4181870">all</span> <span m="4182020">22</span> <span m="4182350">departments.</span> <span m="4184210">Well,</span> <span m="4184310">that</span> <span m="4184490">sounds</span> <span m="4184740">pretty</span> <span m="4185069">good.</span> <span m="4185410">How</span> <span m="4185590">could</span> <span m="4185720">they</span> <span m="4185810">be</span> <span m="4185910">discriminating?</span> </p>
<p><span m="4188029">OK.</span> <span m="4188290">So</span> <span m="4188410">the</span> <span m="4188479">question</span> <span m="4188810">for</span> <span m="4188910">you</span> <span m="4189100">guys,</span> <span m="4190810">is</span> <span m="4190979">it</span> <span m="4191109">possible</span> <span m="4191720">that</span> <span m="4191790">both</span> <span m="4192180">sides</span> <span m="4192779">we're</span> <span m="4192930">telling</span> <span m="4193229">the</span> <span m="4193310">truth,</span> <span m="4194790">that</span> <span m="4194920">in</span> <span m="4195020">every</span> <span m="4195280">single</span> <span m="4195570">department</span> <span m="4196690">the</span> <span m="4196800">women</span> <span m="4197130">have</span> <span m="4197270">a</span> <span m="4197320">lower</span> <span m="4197580">acceptance</span> <span m="4198070">rate</span> <span m="4198270">than</span> <span m="4198400">men,</span> <span m="4199610">but</span> <span m="4200070">on</span> <span m="4200220">the</span> <span m="4200290">university</span> <span m="4200780">as</span> <span m="4200900">a</span> <span m="4200960">whole</span> <span m="4202010">the</span> <span m="4202120">women</span> <span m="4202360">are</span> <span m="4202420">higher</span> <span m="4202800">percentage?</span> <span m="4204850">It</span> <span m="4204990">sounds</span> <span m="4205320">like</span> <span m="4205520">it's--</span> <span m="4206510">and</span> <span m="4206880">just</span> <span m="4207240">to</span> <span m="4207360">avoid</span> <span m="4210140">any</span> <span m="4210310">confusion</span> <span m="4210830">here,</span> <span m="4212450">people</span> <span m="4212810">only</span> <span m="4213110">apply</span> <span m="4213340">to</span> <span m="4213430">one</span> <span m="4213590">department</span> <span m="4215270">and</span> <span m="4216600">they're</span> <span m="4216950">only</span> <span m="4217120">one</span> <span m="4217330">sex.</span> <span m="4218490">So you</span> <span m="4218590">can't--</span> <span m="4219510">Carroll</span> <span m="4220260">didn't</span> <span m="4220430">apply.</span> </p>
<p><span m="4221916">[LAUGHTER]</span> </p>
<p><span m="4225620">How</span> <span m="4226090">many</span> <span m="4226300">people</span> <span m="4226640">think</span> <span m="4226980">that</span> <span m="4227080">one</span> <span m="4227260">of</span> <span m="4227310">the</span> <span m="4227390">sides,</span> <span m="4227780">actually,</span> <span m="4228090">when</span> <span m="4228210">they</span> <span m="4228330">look</span> <span m="4228530">at</span> <span m="4228580">the</span> <span m="4228670">studies</span> <span m="4229050">was</span> <span m="4229200">wrong,</span> <span m="4230040">that</span> <span m="4230210">they're</span> <span m="4230380">contradictory?</span> <span m="4233070">Nobody?</span> <span m="4234270">You've</span> <span m="4234525">been</span> <span m="4234780">in</span> <span m="4235010">6 over 2</span> <span m="4235370">too</span> <span m="4235560">long.</span> <span m="4236140">How</span> <span m="4236270">many</span> <span m="4236440">people</span> <span m="4236680">think</span> <span m="4236840">it's</span> <span m="4236990">possible</span> <span m="4237310">that</span> <span m="4237720">both</span> <span m="4237980">sides</span> <span m="4238230">were</span> <span m="4238350">right?</span> <span m="4240310">Yeah.</span> <span m="4241040">All right.</span> <span m="4241440">So</span> <span m="4241540">let's</span> <span m="4241810">see</span> <span m="4242910">how</span> <span m="4243410">this</span> <span m="4243910">works.</span> </p>
<p><span m="4250940">And to</span> <span m="4251000">make</span> <span m="4251200">it</span> <span m="4251320">simple</span> <span m="4251730">I'm</span> <span m="4251860">going</span> <span m="4251990">to</span> <span m="4252040">get</span> <span m="4252260">down</span> <span m="4252500">to</span> <span m="4252580">just</span> <span m="4252960">two</span> <span m="4253150">departments</span> <span m="4254510">rather</span> <span m="4254910">than</span> <span m="4255080">try</span> <span m="4255230">to</span> <span m="4255340">do</span> <span m="4255470">data</span> <span m="4255770">for</span> <span m="4255920">all</span> <span m="4256060">22.</span> <span m="4256500">And</span> <span m="4256590">I'm</span> <span m="4256680">going</span> <span m="4256800">to</span> <span m="4256870">do</span> <span m="4257890">not</span> <span m="4258140">the</span> <span m="4258530">actual</span> <span m="4258810">data</span> <span m="4259350">but</span> <span m="4259460">something</span> <span m="4259770">that's</span> <span m="4259950">represents</span> <span m="4260140">what's</span> <span m="4260631">going on.</span> <span m="4266040">OK.</span> <span m="4266430">So</span> <span m="4266690">we're</span> <span m="4266790">going</span> <span m="4266870">to</span> <span m="4266940">look</span> <span m="4267130">at</span> <span m="4267200">the</span> <span m="4267260">following</span> <span m="4267790">events.</span> <span m="4272210">A</span> <span m="4272470">is</span> <span m="4272640">the</span> <span m="4272810">event</span> <span m="4273400">that</span> <span m="4273630">the</span> <span m="4274530">applicant</span> <span m="4278530">is</span> <span m="4278990">admitted.</span> <span m="4285930">FCS</span> <span m="4287340">is</span> <span m="4287570">the</span> <span m="4287710">event</span> <span m="4288210">that</span> <span m="4288330">the</span> <span m="4289000">applicant</span> <span m="4290960">is</span> <span m="4291740">female</span> <span m="4295130">and</span> <span m="4296760">applying</span> <span m="4297200">to</span> <span m="4297290">CS.</span> </p>
<p><span m="4300420">FEE</span> <span m="4301970">is</span> <span m="4302160">the</span> <span m="4302280">event</span> <span m="4303160">that</span> <span m="4303280">the</span> <span m="4303400">applicant</span> <span m="4303770">is</span> <span m="4303920">female</span> <span m="4305500">and</span> <span m="4305810">applying</span> <span m="4306280">to</span> <span m="4306450">EE.</span> <span m="4309540">MCS</span> <span m="4312530">is</span> <span m="4312900">the</span> <span m="4313090">event the</span> <span m="4313230">applicant</span> <span m="4313770">is</span> <span m="4314000">a</span> <span m="4314060">male</span> <span m="4316560">and</span> <span m="4316880">CS.</span> <span m="4319140">And</span> <span m="4319270">then</span> <span m="4319410">finally</span> <span m="4319810">we</span> <span m="4319960">have</span> <span m="4321220">MEE</span> <span m="4324490">is the</span> <span m="4324900">event</span> <span m="4325310">the</span> <span m="4325540">applicant</span> <span m="4326010">is</span> <span m="4326150">male</span> <span m="4327170">and</span> <span m="4327710">in</span> <span m="4327960">EE.</span> <span m="4328630">So</span> <span m="4328810">we're</span> <span m="4328870">just</span> <span m="4329070">going</span> <span m="4329170">to</span> <span m="4329210">look</span> <span m="4329360">at</span> <span m="4329460">two</span> <span m="4329570">departments</span> <span m="4330110">here</span> <span m="4330980">and</span> <span m="4331760">try</span> <span m="4331980">to</span> <span m="4332090">figure</span> <span m="4332410">out</span> <span m="4332550">if</span> <span m="4333270">it</span> <span m="4333400">can</span> <span m="4333520">happen</span> <span m="4333950">that</span> <span m="4334080">in</span> <span m="4334180">both</span> <span m="4334500">departments</span> <span m="4335630">the</span> <span m="4335740">women</span> <span m="4335980">are</span> <span m="4336070">worse</span> <span m="4336380">off</span> <span m="4336760">but</span> <span m="4336900">if</span> <span m="4337010">you</span> <span m="4337100">take</span> <span m="4337350">the</span> <span m="4337470">union</span> <span m="4337870">they're</span> <span m="4338270">better</span> <span m="4338670">off.</span> </p>
<p><span m="4345840">So</span> <span m="4347620">the</span> <span m="4347730">female</span> <span m="4348100">professor's</span> <span m="4348650">argument</span> <span m="4349380">effectively</span> <span m="4350070">is,</span> <span m="4351370">the</span> <span m="4351490">probability</span> <span m="4352100">of</span> <span m="4352160">being</span> <span m="4352400">admitted</span> <span m="4353550">given</span> <span m="4353790">that</span> <span m="4353870">you're</span> <span m="4354030">a</span> <span m="4354070">female</span> <span m="4354590">in</span> <span m="4354700">CS</span> <span m="4356940">is</span> <span m="4357180">less</span> <span m="4357570">than</span> <span m="4357650">the</span> <span m="4357720">probability</span> <span m="4358300">of</span> <span m="4358370">being</span> <span m="4358610">admitted</span> <span m="4359730">given</span> <span m="4359980">that you're</span> <span m="4360090">a</span> <span m="4360395">male</span> <span m="4360700">at</span> <span m="4360810">CS.</span> <span m="4363080">And</span> <span m="4364980">same</span> <span m="4365290">thing</span> <span m="4365460">in EE.</span> <span m="4366820">Probability</span> <span m="4367490">of</span> <span m="4367550">being</span> <span m="4367730">admitted</span> <span m="4368060">in</span> <span m="4368180">EE</span> <span m="4368850">if</span> <span m="4369590">you're a</span> <span m="4369910">female</span> <span m="4370210">is</span> <span m="4370510">less</span> <span m="4370760">than</span> <span m="4370860">if</span> <span m="4370970">you're</span> <span m="4371080">a</span> <span m="4371140">male.</span> </p>
<p><span m="4376290">OK?</span> <span m="4377680">Now</span> <span m="4379650">Berkeley</span> <span m="4380170">is</span> <span m="4380240">saying</span> <span m="4383340">it's</span> <span m="4383460">sort of</span> <span m="4383660">the</span> <span m="4383750">reverse.</span> <span m="4384220">The</span> <span m="4384300">probability</span> <span m="4386000">that</span> <span m="4386140">you're</span> <span m="4386300">admitted</span> <span m="4387740">given</span> <span m="4387990">that you're a</span> <span m="4388210">female</span> <span m="4388700">in either</span> <span m="4388990">department</span> <span m="4393260">is</span> <span m="4393490">bigger</span> <span m="4394890">than</span> <span m="4395080">the</span> <span m="4395170">probability</span> <span m="4395770">of</span> <span m="4395840">being</span> <span m="4396010">admitted</span> <span m="4397160">if</span> <span m="4397350">you're</span> <span m="4397530">a</span> <span m="4397570">male</span> <span m="4398860">in</span> <span m="4399000">either</span> <span m="4399230">department.</span> <span m="4404930">OK.</span> <span m="4405220">So</span> <span m="4405360">we've</span> <span m="4405500">now</span> <span m="4406390">expressed</span> <span m="4406960">their</span> <span m="4407160">arguments</span> <span m="4408180">as</span> <span m="4408400">conditional</span> <span m="4408900">probabilities</span> <span m="4411210">Any</span> <span m="4411350">questions?</span> </p>
<p><span m="4413310">Can</span> <span m="4413700">you</span> <span m="4413820">sort</span> <span m="4414040">of</span> <span m="4414100">see</span> <span m="4414430">why</span> <span m="4416130">this</span> <span m="4416320">seems</span> <span m="4416590">contradictory?</span> <span m="4419610">Not</span> <span m="4419780">plus,</span> <span m="4420130">union.</span> <span m="4421800">Because</span> <span m="4422550">this</span> <span m="4422970">is</span> <span m="4423100">sort</span> <span m="4423460">of</span> <span m="4423580">like--</span> <span m="4425190">these</span> <span m="4425430">are</span> <span m="4425490">just</span> <span m="4425690">joined.</span> <span m="4425990">This</span> <span m="4426425">is</span> <span m="4427300">the</span> <span m="4427410">sum</span> <span m="4427660">of</span> <span m="4427710">those.</span> <span m="4430220">And</span> <span m="4430370">this</span> <span m="4430550">is</span> <span m="4430630">sort</span> <span m="4431000">of</span> <span m="4431800">the</span> <span m="4431910">sum</span> <span m="4432140">of</span> <span m="4432210">those.</span> <span m="4434680">And</span> <span m="4434870">yet</span> <span m="4436070">the</span> <span m="4436190">inequality</span> <span m="4436720">changed.</span> </p>
<p><span m="4439150">All</span> <span m="4439210">right.</span> <span m="4439400">In</span> <span m="4439470">fact,</span> <span m="4439740">this</span> <span m="4440000">is</span> <span m="4440540">the</span> <span m="4440680">logic</span> <span m="4442030">that</span> <span m="4442120">we've</span> <span m="4442300">just</span> <span m="4442540">debunked</span> <span m="4442980">over</span> <span m="4443150">there--</span> <span m="4444330">exactly</span> <span m="4444780">that</span> <span m="4444960">claim.</span> <span m="4446306">In</span> <span m="4446800">fact,</span> <span m="4447040">these</span> <span m="4447200">are</span> <span m="4447290">not</span> <span m="4447560">equal</span> <span m="4448970">as</span> <span m="4449150">the</span> <span m="4449260">sum.</span> <span m="4453230">So</span> <span m="4453480">let's</span> <span m="4453660">do</span> <span m="4453790">an</span> <span m="4453860">example.</span> <span m="4458370">Say</span> <span m="4458650">that--</span> <span m="4462450">let's</span> <span m="4462660">do</span> <span m="4462810">it</span> <span m="4462880">over</span> <span m="4463060">here.</span> <span m="4463200">I'll</span> <span m="4463280">put</span> <span m="4463450">the</span> <span m="4463530">real</span> <span m="4463760">values</span> <span m="4464200">in</span> <span m="4464290">over</span> <span m="4464440">here.</span> </p>
<p><span m="4464690">Say</span> <span m="4464940">that</span> <span m="4465170">for</span> <span m="4465320">women</span> <span m="4465630">in</span> <span m="4465680">computer</span> <span m="4466030">science,</span> <span m="4467350">0</span> <span m="4467780">out</span> <span m="4467910">of</span> <span m="4468010">1 were</span> <span m="4468330">admitted</span> <span m="4470020">compared</span> <span m="4470380">to</span> <span m="4470420">the</span> <span m="4470520">men,</span> <span m="4470890">were</span> <span m="4471020">50</span> <span m="4471450">out</span> <span m="4471550">of</span> <span m="4471620">100</span> <span m="4472040">were admitted.</span> <span m="4474800">And</span> <span m="4475040">then</span> <span m="4475160">in</span> <span m="4475270">EE,</span> <span m="4476280">70</span> <span m="4476850">out</span> <span m="4476970">of</span> <span m="4477030">100</span> <span m="4477380">women</span> <span m="4477680">were</span> <span m="4477830">admitted</span> <span m="4479680">compared</span> <span m="4480180">to</span> <span m="4480240">the</span> <span m="4480320">men,</span> <span m="4480650">which</span> <span m="4480830">had</span> <span m="4480990">1</span> <span m="4481270">out</span> <span m="4481380">of</span> <span m="4481480">1.</span> <span m="4484030">All</span> <span m="4484130">right?</span> <span m="4484350">So</span> <span m="4484460">as</span> <span m="4484600">ratios,</span> <span m="4485710">70%</span> <span m="4486490">is less</span> <span m="4486660">than</span> <span m="4486740">100%.</span> <span m="4487450">0%</span> <span m="4488100">is less</span> <span m="4488270">than</span> <span m="4488370">50.</span> </p>
<p><span m="4489730">Now</span> <span m="4489860">if</span> <span m="4489950">I</span> <span m="4490000">look</span> <span m="4490200">at</span> <span m="4490260">the</span> <span m="4490370">two</span> <span m="4490500">departments</span> <span m="4490930">is</span> <span m="4491060">a</span> <span m="4491120">whole,</span> <span m="4491620">I</span> <span m="4491680">get</span> <span m="4491900">70</span> <span m="4493190">over</span> <span m="4493440">101</span> <span m="4495330">is</span> <span m="4495640">in fact</span> <span m="4495930">bigger</span> <span m="4496220">than</span> <span m="4496390">51</span> <span m="4497890">over</span> <span m="4498110">101.</span> <span m="4501140">All right?</span> <span m="4501330">And</span> <span m="4501440">so</span> <span m="4501580">as a whole</span> <span m="4502080">women</span> <span m="4502320">are</span> <span m="4502400">a</span> <span m="4502450">lot</span> <span m="4502670">more</span> <span m="4502790">likely</span> <span m="4503050">to</span> <span m="4503120">be</span> <span m="4503250">admitted</span> <span m="4503630">even</span> <span m="4503830">though in</span> <span m="4503980">each</span> <span m="4504120">department</span> <span m="4504590">they're</span> <span m="4504700">less</span> <span m="4504950">likely</span> <span m="4505220">to</span> <span m="4505280">be</span> <span m="4505410">admitted.</span> <span m="4508020">OK?</span> </p>
<p><span m="4510800">So</span> <span m="4511530">what</span> <span m="4511830">went</span> <span m="4512000">wrong</span> <span m="4512230">with</span> <span m="4512330">the</span> <span m="4512430">intuition,</span> <span m="4513510">which</span> <span m="4513700">you</span> <span m="4513780">didn't</span> <span m="4514080">fall</span> <span m="4514270">victim</span> <span m="4514620">to,</span> <span m="4515320">but</span> <span m="4515490">people</span> <span m="4515790">often</span> <span m="4516170">do,</span> <span m="4517120">that</span> <span m="4517290">it</span> <span m="4518060">shouldn't</span> <span m="4518440">have</span> <span m="4518580">been</span> <span m="4519840">possible</span> <span m="4520310">given</span> <span m="4520560">that?</span> <span m="4521280">What's</span> <span m="4521530">going</span> <span m="4521760">on</span> <span m="4522020">here</span> <span m="4522260">that</span> <span m="4522410">make</span> <span m="4522630">it so that</span> <span m="4523920">it's</span> <span m="4524210">not</span> <span m="4524410">a</span> <span m="4524480">less</span> <span m="4524750">than</span> <span m="4526100">when</span> <span m="4526180">you</span> <span m="4526300">look</span> <span m="4526440">at</span> <span m="4526500">the</span> <span m="4526600">union</span> <span m="4526900">of the departments?</span> <span m="4533440">Yeah?</span> </p>
<p><span m="4534398">AUDIENCE: [INAUDIBLE]</span> <span m="4535294">they're</span> <span m="4535742">weighted differently?</span> </p>
<p><span m="4536190">PROFESSOR: Yeah.</span> <span m="4536640">They're</span> <span m="4537020">weighted</span> <span m="4537200">very</span> <span m="4537610">differently.</span> <span m="4539497">You</span> <span m="4539924">got huge</span> <span m="4540351">waves</span> <span m="4540780">here.</span> <span m="4541740">Right?</span> <span m="4542030">So</span> <span m="4542120">if</span> <span m="4542220">I</span> <span m="4542280">look</span> <span m="4542540">at</span> <span m="4542620">the</span> <span m="4542800">average</span> <span m="4543320">of the</span> <span m="4543440">percentages</span> <span m="4544120">here,</span> <span m="4545740">well</span> <span m="4546030">it's</span> <span m="4546320">35%</span> <span m="4547330">for the</span> <span m="4547440">women</span> <span m="4548350">versus</span> <span m="4548700">75%</span> <span m="4549710">for</span> <span m="4549780">the</span> <span m="4549870">men.</span> <span m="4550960">So</span> <span m="4551070">the</span> <span m="4551280">average</span> <span m="4551720">of the</span> <span m="4551830">percentage</span> <span m="4552380">is</span> <span m="4553160">just</span> <span m="4553390">what</span> <span m="4553460">you'd</span> <span m="4553570">think.</span> <span m="4553860">35</span> <span m="4554440">is</span> <span m="4554560">less</span> <span m="4554770">than</span> <span m="4554890">75.</span> </p>
<p><span m="4556280">But</span> <span m="4556480">I've</span> <span m="4556570">got</span> <span m="4556750">huge</span> <span m="4557220">weightings</span> <span m="4558260">on</span> <span m="4558490">these</span> <span m="4558740">guys,</span> <span m="4559120">which</span> <span m="4559480">changes</span> <span m="4559910">the</span> <span m="4560000">numbers</span> <span m="4561030">quite</span> <span m="4561280">dramatically.</span> <span m="4563180">So</span> <span m="4563300">it all depends</span> <span m="4563792">how you</span> <span m="4564284">count it.</span> <span m="4568720">Actually,</span> <span m="4569040">who</span> <span m="4569190">do</span> <span m="4569240">you</span> <span m="4569310">think</span> <span m="4569620">had</span> <span m="4569790">a</span> <span m="4569830">better--</span> <span m="4570200">Yeah.</span> <span m="4570390">Go</span> <span m="4570490">ahead.</span> </p>
<p><span m="4571226">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="4572120">PROFESSOR: Who</span> <span m="4572260">won</span> <span m="4572400">the</span> <span m="4572480">lawsuit?</span> <span m="4572900">Actually,</span> <span m="4573180">the</span> <span m="4573280">woman</span> <span m="4573530">won</span> <span m="4573680">the</span> <span m="4573760">lawsuit.</span> <span m="4575910">And</span> <span m="4577130">which</span> <span m="4577390">argument</span> <span m="4577710">would</span> <span m="4577840">you</span> <span m="4577920">buy</span> <span m="4578200">now?</span> <span m="4578350">You've</span> <span m="4578470">got</span> <span m="4578620">two</span> <span m="4578800">arguments.</span> <span m="4579220">Which</span> <span m="4579290">one</span> <span m="4579480">would</span> <span m="4579610">you</span> <span m="4579700">believe</span> <span m="4580510">if</span> <span m="4580670">either?</span> <span m="4583120">Which</span> <span m="4583320">one?</span> <span m="4584250">I mean,</span> <span m="4584420">now if</span> <span m="4584660">I look</span> <span m="4584820">at</span> <span m="4584860">exactly</span> <span m="4585340">this</span> <span m="4585570">data</span> <span m="4588820">I</span> <span m="4588880">might</span> <span m="4589150">side--</span> <span m="4590900">I</span> <span m="4590980">might</span> <span m="4591250">side</span> <span m="4591530">with</span> <span m="4591680">Berkeley</span> <span m="4592160">looking</span> <span m="4592410">at</span> <span m="4592500">these</span> <span m="4592710">numbers.</span> </p>
<p><span m="4594400">Then</span> <span m="4594430">again,</span> <span m="4594660">when you</span> <span m="4594760">think</span> <span m="4594980">about</span> <span m="4595230">all</span> <span m="4595510">22</span> <span m="4595940">departments</span> <span m="4597640">and</span> <span m="4597870">the</span> <span m="4597940">fact</span> <span m="4598190">they</span> <span m="4598280">weren't</span> <span m="4598680">this</span> <span m="4598940">lopsided,</span> <span m="4600180">not</span> <span m="4600520">so</span> <span m="4600660">good.</span> <span m="4600940">So</span> <span m="4601130">in</span> <span m="4601200">the</span> <span m="4601340">end</span> <span m="4602210">Berkeley</span> <span m="4602540">lost.</span> <span m="4604050">I'm</span> <span m="4604090">going</span> <span m="4604170">to</span> <span m="4604220">see</span> <span m="4604320">another</span> <span m="4604570">example</span> <span m="4604950">in a</span> <span m="4604990">minute</span> <span m="4605240">where</span> <span m="4605350">it's</span> <span m="4605450">even</span> <span m="4605630">more</span> <span m="4605940">clear</span> <span m="4606220">which</span> <span m="4606430">side</span> <span m="4606920">to</span> <span m="4606970">believe</span> <span m="4607280">in.</span> <span m="4607360">But</span> <span m="4607460">it</span> <span m="4607520">really</span> <span m="4607730">depends</span> <span m="4608200">on</span> <span m="4608820">the</span> <span m="4608920">numbers</span> <span m="4609330">as</span> <span m="4609430">to</span> <span m="4609520">which</span> <span m="4609690">one</span> <span m="4609870">you</span> <span m="4609980">might,</span> <span m="4610880">if you had</span> <span m="4611100">to</span> <span m="4611140">vote,</span> <span m="4611460">which</span> <span m="4611590">way</span> <span m="4611730">you'd</span> <span m="4611850">vote.</span> </p>
<p><span m="4614070">Here's</span> <span m="4614490">another</span> <span m="4614810">example.</span> <span m="4616390">This</span> <span m="4616530">is</span> <span m="4616650">from</span> <span m="4616770">a</span> <span m="4616810">newspaper</span> <span m="4617380">article</span> <span m="4618440">on</span> <span m="4618780">which</span> <span m="4619090">airlines</span> <span m="4620280">are</span> <span m="4620440">best</span> <span m="4620820">to</span> <span m="4620900">fly</span> <span m="4621380">because</span> <span m="4621800">they</span> <span m="4621910">have</span> <span m="4622060">the</span> <span m="4622523">best on-time</span> <span m="4622986">rates.</span> <span m="4625764">And</span> <span m="4627160">in</span> <span m="4627480">this</span> <span m="4627690">case</span> <span m="4627950">they</span> <span m="4628080">were</span> <span m="4628760">comparing</span> <span m="4629320">American</span> <span m="4629750">Airlines</span> <span m="4630300">and</span> <span m="4630390">America</span> <span m="4630820">West,</span> <span m="4632310">looking</span> <span m="4632590">at</span> <span m="4632660">on-time</span> <span m="4633260">rates.</span> <span m="4634850">And</span> <span m="4635070">here's</span> <span m="4635370">the</span> <span m="4635470">data</span> <span m="4635760">they</span> <span m="4635880">showed</span> <span m="4636225">for</span> <span m="4636570">the</span> <span m="4636680">two</span> <span m="4636880">airlines.</span> </p>
<p><span m="4638550">Here's</span> <span m="4638740">American</span> <span m="4639170">Airlines.</span> <span m="4640520">Here's</span> <span m="4640730">America</span> <span m="4641170">West.</span> <span m="4643510">And</span> <span m="4643720">they</span> <span m="4644110">took</span> <span m="4644450">five</span> <span m="4645000">cities,</span> <span m="4647020">LA,</span> <span m="4647690">Phoenix,</span> <span m="4649500">San</span> <span m="4649800">Diego,</span> <span m="4651070">San</span> <span m="4651300">Francisco,</span> <span m="4652610">and</span> <span m="4652750">Seattle.</span> <span m="4656510">And</span> <span m="4656880">then</span> <span m="4657010">you</span> <span m="4657160">looked</span> <span m="4657390">at</span> <span m="4657530">the</span> <span m="4657620">number</span> <span m="4657940">on</span> <span m="4658160">time,</span> <span m="4661350">the</span> <span m="4661440">number</span> <span m="4661770">of</span> <span m="4661850">flights,</span> <span m="4663230">and</span> <span m="4663360">then</span> <span m="4663490">the</span> <span m="4663630">rate,</span> <span m="4665080">percentage</span> <span m="4665980">on</span> <span m="4666150">time.</span> <span m="4667420">And</span> <span m="4667630">then</span> <span m="4667810">same</span> <span m="4668040">thing</span> <span m="4668200">here.</span> <span m="4668390">Number</span> <span m="4668690">on</span> <span m="4668880">time,</span> <span m="4672640">number</span> <span m="4672880">of</span> <span m="4672940">flights,</span> <span m="4674230">and</span> <span m="4674390">the</span> <span m="4674480">rate.</span> </p>
<p><span m="4677300">So</span> <span m="4677410">I'm</span> <span m="4677470">just</span> <span m="4677620">going</span> <span m="4677720">to</span> <span m="4677790">give</span> <span m="4677950">you</span> <span m="4678000">the</span> <span m="4678090">numbers</span> <span m="4678500">here.</span> <span m="4679800">So</span> <span m="4680020">they</span> <span m="4680160">had</span> <span m="4680430">500</span> <span m="4682300">out</span> <span m="4682540">of</span> <span m="4683040">560</span> <span m="4685280">for</span> <span m="4685460">a rate</span> <span m="4685900">of</span> <span m="4686362">89%,</span> <span m="4690520">220</span> <span m="4692280">over</span> <span m="4692620">230</span> <span m="4694100">for</span> <span m="4694190">95,</span> <span m="4698160">210</span> <span m="4698380">over</span> <span m="4699830">230</span> <span m="4701036">for</span> <span m="4701440">92%,</span> <span m="4705000">500</span> <span m="4706260">over</span> <span m="4706540">600</span> <span m="4708304">for</span> <span m="4708745">83%,</span> <span m="4711570">and</span> <span m="4711700">then</span> <span m="4711840">Seattle.</span> <span m="4712430">They</span> <span m="4712520">had a</span> <span m="4712570">lot</span> <span m="4712860">of</span> <span m="4712940">flights.</span> <span m="4713580">That's</span> <span m="4713820">where</span> <span m="4714210">they're--</span> <span m="4714930">we have</span> <span m="4715260">a</span> <span m="4715350">hub</span> <span m="4716060">of</span> <span m="4716700">2,200</span> <span m="4718930">for</span> <span m="4719380">86%.</span> </p>
<p><span m="4720970">And</span> <span m="4721130">if</span> <span m="4721200">you</span> <span m="4721280">added</span> <span m="4721510">them</span> <span m="4721720">all</span> <span m="4722080">up,</span> <span m="4724900">they</span> <span m="4724940">got</span> <span m="4725640">3,300</span> <span m="4727560">out</span> <span m="4727690">of</span> <span m="4728000">3,820</span> <span m="4728520">for</span> <span m="4730470">87%</span> <span m="4732610">on</span> <span m="4732820">time.</span> <span m="4734260">Now</span> <span m="4734580">the</span> <span m="4734690">data</span> <span m="4735020">for</span> <span m="4735180">American</span> <span m="4735600">West</span> <span m="4736110">looks</span> <span m="4736260">something</span> <span m="4736480">like</span> <span m="4736650">the</span> <span m="4736720">following.</span> <span m="4738150">In</span> <span m="4738300">LA</span> <span m="4738880">it's</span> <span m="4739040">700</span> <span m="4739535">out of</span> <span m="4740030">800</span> <span m="4741020">for</span> <span m="4742292">87%.</span> <span m="4746170">they're</span> <span m="4746430">based</span> <span m="4746770">in</span> <span m="4746860">Phoenix.</span> <span m="4747470">They</span> <span m="4747550">got</span> <span m="4747690">a</span> <span m="4747870">zillion</span> <span m="4748250">flights</span> <span m="4748620">there.</span> <span m="4748860">4,900</span> <span m="4750213">out of</span> <span m="4750664">5,300</span> <span m="4752017">for</span> <span m="4752919">92%.</span> <span m="4756540">And</span> <span m="4756730">400</span> <span m="4757724">over</span> <span m="4758720">450</span> <span m="4760320">for</span> <span m="4760510">89%,</span> <span m="4763420">320,</span> <span m="4764884">over</span> <span m="4765372">450,</span> <span m="4766630">71%,</span> <span m="4769280">200</span> <span m="4770710">over</span> <span m="4770940">260</span> <span m="4771860">for</span> <span m="4772540">77%.</span> <span m="4774290">And</span> <span m="4774400">then</span> <span m="4774520">you</span> <span m="4774610">add</span> <span m="4774810">all</span> <span m="4775020">them</span> <span m="4775250">up.</span> <span m="4775730">And</span> <span m="4775850">you've</span> <span m="4775960">got</span> <span m="4776670">6,520</span> <span m="4778740">over</span> <span m="4779750">7,260</span> <span m="4781896">for</span> <span m="4782368">90%.</span> </p>
<p><span m="4785110">So</span> <span m="4785260">the</span> <span m="4785360">newspaper</span> <span m="4785970">concluded</span> <span m="4786790">and</span> <span m="4786990">literally</span> <span m="4787290">said</span> <span m="4788530">that</span> <span m="4788650">American</span> <span m="4789130">West</span> <span m="4789580">is</span> <span m="4789680">the</span> <span m="4789760">better</span> <span m="4790070">airline</span> <span m="4790510">to</span> <span m="4790580">fly</span> <span m="4791540">because</span> <span m="4791800">they're</span> <span m="4791940">on-time</span> <span m="4792510">rate</span> <span m="4792750">is</span> <span m="4792870">much</span> <span m="4793110">better.</span> <span m="4793410">It's</span> <span m="4793570">90%</span> <span m="4794860">versus</span> <span m="4795540">87%.</span> <span m="4798880">What</span> <span m="4799000">do</span> <span m="4799040">you</span> <span m="4799140">think?</span> <span m="4800420">Which</span> <span m="4800660">airline</span> <span m="4801110">would you</span> <span m="4801200">fly</span> <span m="4801510">looking</span> <span m="4801985">at</span> <span m="4802460">that data?</span> </p>
<p><span m="4804360">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="4810550">PROFESSOR: I</span> <span m="4810660">know</span> <span m="4810940">which</span> <span m="4811090">one I'd</span> <span m="4811180">fly.</span> <span m="4813890">It</span> <span m="4814240">looks like</span> <span m="4814430">America</span> <span m="4814820">West</span> <span m="4815090">is</span> <span m="4815170">better.</span> <span m="4816790">Every</span> <span m="4817330">single</span> <span m="4817740">city,</span> <span m="4819400">American</span> <span m="4819750">Airlines</span> <span m="4820160">is</span> <span m="4820250">better.</span> <span m="4823410">92</span> <span m="4824320">versus</span> <span m="4824780">89.</span> <span m="4825380">Everywhere</span> <span m="4826190">it's better</span> <span m="4826400">by a bunch.</span> <span m="4826890">83</span> <span m="4827380">versus</span> <span m="4827700">71.</span> <span m="4829010">86</span> <span m="4829660">versus</span> <span m="4829970">77.</span> <span m="4831330">Every</span> <span m="4831670">single</span> <span m="4832050">city,</span> <span m="4833360">American</span> <span m="4833720">Airlines</span> <span m="4834110">is</span> <span m="4834230">better.</span> <span m="4836510">Yet,</span> <span m="4836810">America</span> <span m="4837170">West</span> <span m="4837440">is</span> <span m="4837530">better</span> <span m="4837770">overall.</span> <span m="4840480">And</span> <span m="4840560">that's</span> <span m="4840720">what</span> <span m="4840810">the</span> <span m="4840880">newspaper</span> <span m="4841340">said.</span> <span m="4841880">They</span> <span m="4841980">went</span> <span m="4842140">on</span> <span m="4842250">this.</span> </p>
<p><span m="4843480">But</span> <span m="4843670">of</span> <span m="4843730">course,</span> <span m="4844070">no</span> <span m="4844130">matter</span> <span m="4844330">where</span> <span m="4844530">you're</span> <span m="4844835">going you're</span> <span m="4845140">better</span> <span m="4845390">off</span> <span m="4845560">with</span> <span m="4845660">American</span> <span m="4846010">Airlines.</span> <span m="4849060">All right?</span> <span m="4849460">Now</span> <span m="4851624">what happened</span> <span m="4852075">here?</span> <span m="4853880">The</span> <span m="4854000">waiting.</span> <span m="4855270">In</span> <span m="4855400">fact,</span> <span m="4857440">America</span> <span m="4857920">West</span> <span m="4858440">flies</span> <span m="4858810">out</span> <span m="4858920">of</span> <span m="4858990">Phoenix</span> <span m="4859460">where</span> <span m="4859560">the</span> <span m="4859670">weather's</span> <span m="4860130">great.</span> <span m="4862520">So</span> <span m="4862710">you</span> <span m="4862850">get</span> <span m="4863150">a</span> <span m="4863210">higher</span> <span m="4863660">on-time</span> <span m="4864180">rate</span> <span m="4864380">when</span> <span m="4864840">in</span> <span m="4864970">a</span> <span m="4865020">good-weather</span> <span m="4865460">city.</span> <span m="4865740">And</span> <span m="4865800">they</span> <span m="4865880">got</span> <span m="4866060">most</span> <span m="4866350">of their</span> <span m="4866480">flights</span> <span m="4866800">there.</span> <span m="4867690">American</span> <span m="4868110">Airlines</span> <span m="4868820">got</span> <span m="4868940">a lot</span> <span m="4869350">of flights in</span> <span m="4869500">Seattle</span> <span m="4869940">where</span> <span m="4870040">the</span> <span m="4870140">weather</span> <span m="4870800">sucks</span> <span m="4872120">and</span> <span m="4872480">you're</span> <span m="4872600">always</span> <span m="4872920">delayed.</span> </p>
<p><span m="4874580">All</span> <span m="4874660">right?</span> <span m="4874890">And</span> <span m="4875010">so</span> <span m="4875240">they</span> <span m="4875450">look</span> <span m="4875690">worse</span> <span m="4876050">on</span> <span m="4876190">average</span> <span m="4876530">because</span> <span m="4876680">so</span> <span m="4876820">many</span> <span m="4876990">of their</span> <span m="4877130">flights</span> <span m="4877430">are</span> <span m="4877610">in a</span> <span m="4877660">bad</span> <span m="4877970">city</span> <span m="4879065">and</span> <span m="4879350">so</span> <span m="4879520">many</span> <span m="4879660">of</span> <span m="4879720">America</span> <span m="4880120">West</span> <span m="4880550">are in a</span> <span m="4880600">good</span> <span m="4880800">city.</span> <span m="4882900">All</span> <span m="4883010">right?</span> <span m="4883250">So</span> <span m="4883450">it</span> <span m="4883550">makes</span> <span m="4883860">America</span> <span m="4884140">West</span> <span m="4884450">look</span> <span m="4884630">better</span> <span m="4884950">when</span> <span m="4885080">in</span> <span m="4885170">fact,</span> <span m="4885700">in</span> <span m="4885760">this</span> <span m="4885920">case,</span> <span m="4886230">it's</span> <span m="4886250">absolutely</span> <span m="4886770">clear</span> <span m="4887060">whose</span> <span m="4887290">better.</span> </p>
<p><span m="4888420">American</span> <span m="4888940">Airlines</span> <span m="4889320">is</span> <span m="4889440">better,</span> <span m="4889740">every</span> <span m="4890040">single</span> <span m="4890330">city.</span> <span m="4891422">All right.</span> <span m="4892030">That's</span> <span m="4892330">why</span> <span m="4892680">Mark</span> <span m="4893000">Twain</span> <span m="4893340">said,</span> <span m="4894690">"There's</span> <span m="4894860">three</span> <span m="4895050">kinds</span> <span m="4895370">of</span> <span m="4895450">lies--</span> <span m="4896390">lies,</span> <span m="4897360">damned</span> <span m="4897590">lies,</span> <span m="4898300">and</span> <span m="4898340">statistics."</span> <span m="4899700">We'll</span> <span m="4899800">see</span> <span m="4899960">more</span> <span m="4900260">examples</span> <span m="4900560">next</span> <span m="4900860">time.</span> </p>
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